Classical Invariant Theory¶
This module lists classical invariants and covariants of homogeneous polynomials (also called algebraic forms) under the action of the special linear group. That is, we are dealing with polynomials of degree \(d\) in \(n\) variables. The special linear group \(SL(n,\CC)\) acts on the variables \((x_1,\dots, x_n)\) linearly,
\[(x_1,\dots, x_n)^t \to A (x_1,\dots, x_n)^t
,\qquad
A \in SL(n,\CC)\]
The linear action on the variables transforms a polynomial \(p\) generally into a different polynomial \(gp\). We can think of it as an action on the space of coefficients in \(p\). An invariant is a polynomial in the coefficients that is invariant under this action. A covariant is a polynomial in the coefficients and the variables \((x_1,\dots, x_n)\) that is invariant under the combined action.
For example, the binary quadratic \(p(x,y) = a x^2 + b x y + c y^2\) has as its invariant the discriminant \(\mathop{disc}(p) = b^2 - 4 a c\). This means that for any \(SL(2,\CC)\) coordinate change
\[\begin{split}\begin{pmatrix} x' \\ y' \end{pmatrix}
=
\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}
\qquad
\alpha\delta-\beta\gamma=1\end{split}\]
the discriminant is invariant, \(\mathop{disc}\big(p(x',y')\big) = \mathop{disc}\big(p(x,y)\big)\).
To use this module, you should use the factory object
invariant_theory. For example, take
the quartic:
sage: R.<x,y> = QQ[]
sage: q = x^4 + y^4
sage: quartic = invariant_theory.binary_quartic(q); quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)
>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> q = x**Integer(4) + y**Integer(4)
>>> quartic = invariant_theory.binary_quartic(q); quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)
R.<x,y> = QQ[] q = x^4 + y^4 quartic = invariant_theory.binary_quartic(q); quartic
One invariant of a quartic is known as the Eisenstein D-invariant. Since it is an invariant, it is a polynomial in the coefficients (which are integers in this example):
sage: quartic.EisensteinD()
1
>>> from sage.all import *
>>> quartic.EisensteinD()
1
quartic.EisensteinD()
One example of a covariant of a quartic is the so-called g-covariant (actually, the Hessian). As with all covariants, it is a polynomial in \(x\), \(y\) and the coefficients:
sage: quartic.g_covariant()
-x^2*y^2
>>> from sage.all import *
>>> quartic.g_covariant()
-x^2*y^2
quartic.g_covariant()
As usual, use tab completion and the online help to discover the implemented invariants and covariants.
In general, the variables of the defining polynomial cannot be guessed. For example, the zero polynomial can be thought of as a homogeneous polynomial of any degree. Also, since we also want to allow polynomial coefficients we cannot just take all variables of the polynomial ring as the variables of the form. This is why you will have to specify the variables explicitly if there is any potential ambiguity. For example:
sage: invariant_theory.binary_quartic(R.zero(), [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 0)
sage: invariant_theory.binary_quartic(x^4, [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 1)
sage: R.<x,y,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y])
Binary quartic with coefficients (1, 0, t, 0, 1)
>>> from sage.all import *
>>> invariant_theory.binary_quartic(R.zero(), [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 0)
>>> invariant_theory.binary_quartic(x**Integer(4), [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 1)
>>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3)
>>> invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4) + t*x**Integer(2)*y**Integer(2), [x,y])
Binary quartic with coefficients (1, 0, t, 0, 1)
invariant_theory.binary_quartic(R.zero(), [x,y]) invariant_theory.binary_quartic(x^4, [x,y]) R.<x,y,t> = QQ[] invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y])
Finally, it is often convenient to use inhomogeneous polynomials where it is understood that one wants to homogenize them. This is also supported, just define the form with an inhomogeneous polynomial and specify one less variable:
sage: R.<x,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x])
Binary quartic with coefficients (1, 0, t, 0, 1)
>>> from sage.all import *
>>> R = QQ['x, t']; (x, t,) = R._first_ngens(2)
>>> invariant_theory.binary_quartic(x**Integer(4) + Integer(1) + t*x**Integer(2), [x])
Binary quartic with coefficients (1, 0, t, 0, 1)
R.<x,t> = QQ[] invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x])
REFERENCES:
AUTHORS:
Volker Braun (2013-01-24): initial version
Jesper Noordsij (2018-05-18): support for binary quintics added
- class sage.rings.invariants.invariant_theory.AlgebraicForm(n, d, polynomial, *args, **kwds)[source]¶
Bases:
FormsBaseThe base class of algebraic forms (i.e. homogeneous polynomials).
You should only instantiate the derived classes of this base class.
Derived classes must implement
coeffs()andscaled_coeffs()INPUT:
n– the number of variablesd– the degree of the polynomialpolynomial– the polynomial*args– the variables, as a single list/tuple, multiple arguments, orNoneto use all variables of the polynomial
Derived classes must implement the same arguments for the constructor.
EXAMPLES:
sage: from sage.rings.invariants.invariant_theory import AlgebraicForm sage: R.<x,y> = QQ[] sage: p = x^2 + y^2 sage: AlgebraicForm(2, 2, p).variables() (x, y) sage: AlgebraicForm(2, 2, p, None).variables() (x, y) sage: AlgebraicForm(3, 2, p).variables() (x, y, None) sage: AlgebraicForm(3, 2, p, None).variables() (x, y, None) sage: from sage.rings.invariants.invariant_theory import AlgebraicForm sage: R.<x,y,s,t> = QQ[] sage: p = s*x^2 + t*y^2 sage: AlgebraicForm(2, 2, p, [x,y]).variables() (x, y) sage: AlgebraicForm(2, 2, p, x,y).variables() (x, y) sage: AlgebraicForm(3, 2, p, [x,y,None]).variables() (x, y, None) sage: AlgebraicForm(3, 2, p, x,y,None).variables() (x, y, None) sage: AlgebraicForm(2, 1, p, [x,y]).variables() Traceback (most recent call last): ... ValueError: polynomial is of the wrong degree sage: AlgebraicForm(2, 2, x^2 + y, [x,y]).variables() Traceback (most recent call last): ... ValueError: polynomial is not homogeneous
>>> from sage.all import * >>> from sage.rings.invariants.invariant_theory import AlgebraicForm >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> p = x**Integer(2) + y**Integer(2) >>> AlgebraicForm(Integer(2), Integer(2), p).variables() (x, y) >>> AlgebraicForm(Integer(2), Integer(2), p, None).variables() (x, y) >>> AlgebraicForm(Integer(3), Integer(2), p).variables() (x, y, None) >>> AlgebraicForm(Integer(3), Integer(2), p, None).variables() (x, y, None) >>> from sage.rings.invariants.invariant_theory import AlgebraicForm >>> R = QQ['x, y, s, t']; (x, y, s, t,) = R._first_ngens(4) >>> p = s*x**Integer(2) + t*y**Integer(2) >>> AlgebraicForm(Integer(2), Integer(2), p, [x,y]).variables() (x, y) >>> AlgebraicForm(Integer(2), Integer(2), p, x,y).variables() (x, y) >>> AlgebraicForm(Integer(3), Integer(2), p, [x,y,None]).variables() (x, y, None) >>> AlgebraicForm(Integer(3), Integer(2), p, x,y,None).variables() (x, y, None) >>> AlgebraicForm(Integer(2), Integer(1), p, [x,y]).variables() Traceback (most recent call last): ... ValueError: polynomial is of the wrong degree >>> AlgebraicForm(Integer(2), Integer(2), x**Integer(2) + y, [x,y]).variables() Traceback (most recent call last): ... ValueError: polynomial is not homogeneous
from sage.rings.invariants.invariant_theory import AlgebraicForm R.<x,y> = QQ[] p = x^2 + y^2 AlgebraicForm(2, 2, p).variables() AlgebraicForm(2, 2, p, None).variables() AlgebraicForm(3, 2, p).variables() AlgebraicForm(3, 2, p, None).variables() from sage.rings.invariants.invariant_theory import AlgebraicForm R.<x,y,s,t> = QQ[] p = s*x^2 + t*y^2 AlgebraicForm(2, 2, p, [x,y]).variables() AlgebraicForm(2, 2, p, x,y).variables() AlgebraicForm(3, 2, p, [x,y,None]).variables() AlgebraicForm(3, 2, p, x,y,None).variables() AlgebraicForm(2, 1, p, [x,y]).variables() AlgebraicForm(2, 2, x^2 + y, [x,y]).variables()
- coefficients()[source]¶
Alias for
coeffs().See the documentation for
coeffs()for details.EXAMPLES:
sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[] sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z sage: q = invariant_theory.quadratic_form(p, x,y,z) sage: q.coefficients() (a, b, c, d, e, f) sage: q.coeffs() (a, b, c, d, e, f)
>>> from sage.all import * >>> R = QQ['a, b, c, d, e, f, g, x, y, z']; (a, b, c, d, e, f, g, x, y, z,) = R._first_ngens(10) >>> p = a*x**Integer(2) + b*y**Integer(2) + c*z**Integer(2) + d*x*y + e*x*z + f*y*z >>> q = invariant_theory.quadratic_form(p, x,y,z) >>> q.coefficients() (a, b, c, d, e, f) >>> q.coeffs() (a, b, c, d, e, f)
R.<a,b,c,d,e,f,g, x,y,z> = QQ[] p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z q = invariant_theory.quadratic_form(p, x,y,z) q.coefficients() q.coeffs()
- form()[source]¶
Return the defining polynomial.
OUTPUT: the polynomial used to define the algebraic form
EXAMPLES:
sage: R.<x,y> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4) sage: quartic.form() x^4 + y^4 sage: quartic.polynomial() x^4 + y^4
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4)) >>> quartic.form() x^4 + y^4 >>> quartic.polynomial() x^4 + y^4
R.<x,y> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4) quartic.form() quartic.polynomial()
- homogenized(var='h')[source]¶
Return form as defined by a homogeneous polynomial.
INPUT:
var– either a variable name, variable index or a variable (default:'h')
OUTPUT:
The same algebraic form, but defined by a homogeneous polynomial.
EXAMPLES:
sage: T.<t> = QQ[] sage: quadratic = invariant_theory.binary_quadratic(t^2 + 2*t + 3) sage: quadratic Binary quadratic with coefficients (1, 3, 2) sage: quadratic.homogenized() Binary quadratic with coefficients (1, 3, 2) sage: quadratic == quadratic.homogenized() True sage: quadratic.form() t^2 + 2*t + 3 sage: quadratic.homogenized().form() t^2 + 2*t*h + 3*h^2 sage: R.<x,y,z> = QQ[] sage: quadratic = invariant_theory.ternary_quadratic(x^2 + 1, [x,y]) sage: quadratic.homogenized().form() x^2 + h^2 sage: R.<x> = QQ[] sage: quintic = invariant_theory.binary_quintic(x^4 + 1, x) sage: quintic.homogenized().form() x^4*h + h^5
>>> from sage.all import * >>> T = QQ['t']; (t,) = T._first_ngens(1) >>> quadratic = invariant_theory.binary_quadratic(t**Integer(2) + Integer(2)*t + Integer(3)) >>> quadratic Binary quadratic with coefficients (1, 3, 2) >>> quadratic.homogenized() Binary quadratic with coefficients (1, 3, 2) >>> quadratic == quadratic.homogenized() True >>> quadratic.form() t^2 + 2*t + 3 >>> quadratic.homogenized().form() t^2 + 2*t*h + 3*h^2 >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quadratic = invariant_theory.ternary_quadratic(x**Integer(2) + Integer(1), [x,y]) >>> quadratic.homogenized().form() x^2 + h^2 >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> quintic = invariant_theory.binary_quintic(x**Integer(4) + Integer(1), x) >>> quintic.homogenized().form() x^4*h + h^5
T.<t> = QQ[] quadratic = invariant_theory.binary_quadratic(t^2 + 2*t + 3) quadratic quadratic.homogenized() quadratic == quadratic.homogenized() quadratic.form() quadratic.homogenized().form() R.<x,y,z> = QQ[] quadratic = invariant_theory.ternary_quadratic(x^2 + 1, [x,y]) quadratic.homogenized().form() R.<x> = QQ[] quintic = invariant_theory.binary_quintic(x^4 + 1, x) quintic.homogenized().form()
- polynomial()[source]¶
Return the defining polynomial.
OUTPUT: the polynomial used to define the algebraic form
EXAMPLES:
sage: R.<x,y> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4) sage: quartic.form() x^4 + y^4 sage: quartic.polynomial() x^4 + y^4
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4)) >>> quartic.form() x^4 + y^4 >>> quartic.polynomial() x^4 + y^4
R.<x,y> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4) quartic.form() quartic.polynomial()
- transformed(g)[source]¶
Return the image under a linear transformation of the variables.
INPUT:
g– a \(GL(n,\CC)\) matrix or a dictionary with the variables as keys. A matrix is used to define the linear transformation of homogeneous variables, a dictionary acts by substitution of the variables.
OUTPUT:
A new instance of a subclass of
AlgebraicFormobtained by replacing the variables of the homogeneous polynomial by their image underg.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + 2*y^3 + 3*z^3 + 4*x*y*z) sage: cubic.transformed({x: y, y: z, z: x}).form() 3*x^3 + y^3 + 4*x*y*z + 2*z^3 sage: cyc = matrix([[0,1,0], [0,0,1], [1,0,0]]) sage: cubic.transformed(cyc) == cubic.transformed({x:y, y:z, z:x}) True sage: g = matrix(QQ, [[1, 0, 0], [-1, 1, -3], [-5, -5, 16]]) sage: cubic.transformed(g) Ternary cubic with coefficients (-356, -373, 12234, -1119, 3578, -1151, 3582, -11766, -11466, 7360) sage: cubic.transformed(g).transformed(g.inverse()) == cubic True
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + Integer(2)*y**Integer(3) + Integer(3)*z**Integer(3) + Integer(4)*x*y*z) >>> cubic.transformed({x: y, y: z, z: x}).form() 3*x^3 + y^3 + 4*x*y*z + 2*z^3 >>> cyc = matrix([[Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)], [Integer(1),Integer(0),Integer(0)]]) >>> cubic.transformed(cyc) == cubic.transformed({x:y, y:z, z:x}) True >>> g = matrix(QQ, [[Integer(1), Integer(0), Integer(0)], [-Integer(1), Integer(1), -Integer(3)], [-Integer(5), -Integer(5), Integer(16)]]) >>> cubic.transformed(g) Ternary cubic with coefficients (-356, -373, 12234, -1119, 3578, -1151, 3582, -11766, -11466, 7360) >>> cubic.transformed(g).transformed(g.inverse()) == cubic True
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + 2*y^3 + 3*z^3 + 4*x*y*z) cubic.transformed({x: y, y: z, z: x}).form() cyc = matrix([[0,1,0], [0,0,1], [1,0,0]]) cubic.transformed(cyc) == cubic.transformed({x:y, y:z, z:x}) g = matrix(QQ, [[1, 0, 0], [-1, 1, -3], [-5, -5, 16]]) cubic.transformed(g) cubic.transformed(g).transformed(g.inverse()) == cubic
- class sage.rings.invariants.invariant_theory.BinaryQuartic(n, d, polynomial, *args)[source]¶
Bases:
AlgebraicFormInvariant theory of a binary quartic.
You should use the
invariant_theoryfactory object to construct instances of this class. Seebinary_quartic()for details.- EisensteinD()[source]¶
One of the Eisenstein invariants of a binary quartic.
OUTPUT: the Eisenstein D-invariant of the quartic
\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = a_0 a_4+3 a_2^2-4 a_1 a_3\end{split}\]EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] sage: f = a0*x1^4 + 4*a1*x0*x1^3 + 6*a2*x0^2*x1^2 + 4*a3*x0^3*x1 + a4*x0^4 sage: inv = invariant_theory.binary_quartic(f, x0, x1) sage: inv.EisensteinD() 3*a2^2 - 4*a1*a3 + a0*a4
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x0, x1']; (a0, a1, a2, a3, a4, x0, x1,) = R._first_ngens(7) >>> f = a0*x1**Integer(4) + Integer(4)*a1*x0*x1**Integer(3) + Integer(6)*a2*x0**Integer(2)*x1**Integer(2) + Integer(4)*a3*x0**Integer(3)*x1 + a4*x0**Integer(4) >>> inv = invariant_theory.binary_quartic(f, x0, x1) >>> inv.EisensteinD() 3*a2^2 - 4*a1*a3 + a0*a4
R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] f = a0*x1^4 + 4*a1*x0*x1^3 + 6*a2*x0^2*x1^2 + 4*a3*x0^3*x1 + a4*x0^4 inv = invariant_theory.binary_quartic(f, x0, x1) inv.EisensteinD()
- EisensteinE()[source]¶
One of the Eisenstein invariants of a binary quartic.
OUTPUT: the Eisenstein E-invariant of the quartic
\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow E(f) = a_0 a_3^2 +a_1^2 a_4 -a_0 a_2 a_4 -2 a_1 a_2 a_3 + a_2^3\end{split}\]EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] sage: f = a0*x1^4 + 4*a1*x0*x1^3 + 6*a2*x0^2*x1^2 + 4*a3*x0^3*x1 + a4*x0^4 sage: inv = invariant_theory.binary_quartic(f, x0, x1) sage: inv.EisensteinE() a2^3 - 2*a1*a2*a3 + a0*a3^2 + a1^2*a4 - a0*a2*a4
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x0, x1']; (a0, a1, a2, a3, a4, x0, x1,) = R._first_ngens(7) >>> f = a0*x1**Integer(4) + Integer(4)*a1*x0*x1**Integer(3) + Integer(6)*a2*x0**Integer(2)*x1**Integer(2) + Integer(4)*a3*x0**Integer(3)*x1 + a4*x0**Integer(4) >>> inv = invariant_theory.binary_quartic(f, x0, x1) >>> inv.EisensteinE() a2^3 - 2*a1*a2*a3 + a0*a3^2 + a1^2*a4 - a0*a2*a4
R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] f = a0*x1^4 + 4*a1*x0*x1^3 + 6*a2*x0^2*x1^2 + 4*a3*x0^3*x1 + a4*x0^4 inv = invariant_theory.binary_quartic(f, x0, x1) inv.EisensteinE()
- coeffs()[source]¶
The coefficients of a binary quartic.
Given
\[f(x) = a_0 x_1^4 + a_1 x_0 x_1^3 + a_2 x_0^2 x_1^2 + a_3 x_0^3 x_1 + a_4 x_0^4\]this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] sage: p = a0*x1^4 + a1*x1^3*x0 + a2*x1^2*x0^2 + a3*x1*x0^3 + a4*x0^4 sage: quartic = invariant_theory.binary_quartic(p, x0, x1) sage: quartic.coeffs() (a0, a1, a2, a3, a4) sage: R.<a0, a1, a2, a3, a4, x> = QQ[] sage: p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 sage: quartic = invariant_theory.binary_quartic(p, x) sage: quartic.coeffs() (a0, a1, a2, a3, a4)
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x0, x1']; (a0, a1, a2, a3, a4, x0, x1,) = R._first_ngens(7) >>> p = a0*x1**Integer(4) + a1*x1**Integer(3)*x0 + a2*x1**Integer(2)*x0**Integer(2) + a3*x1*x0**Integer(3) + a4*x0**Integer(4) >>> quartic = invariant_theory.binary_quartic(p, x0, x1) >>> quartic.coeffs() (a0, a1, a2, a3, a4) >>> R = QQ['a0, a1, a2, a3, a4, x']; (a0, a1, a2, a3, a4, x,) = R._first_ngens(6) >>> p = a0 + a1*x + a2*x**Integer(2) + a3*x**Integer(3) + a4*x**Integer(4) >>> quartic = invariant_theory.binary_quartic(p, x) >>> quartic.coeffs() (a0, a1, a2, a3, a4)
R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] p = a0*x1^4 + a1*x1^3*x0 + a2*x1^2*x0^2 + a3*x1*x0^3 + a4*x0^4 quartic = invariant_theory.binary_quartic(p, x0, x1) quartic.coeffs() R.<a0, a1, a2, a3, a4, x> = QQ[] p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 quartic = invariant_theory.binary_quartic(p, x) quartic.coeffs()
- g_covariant()[source]¶
The g-covariant of a binary quartic.
OUTPUT: the g-covariant of the quartic
\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = \frac{1}{144} \begin{pmatrix} \frac{\partial^2 f}{\partial x \partial x} \end{pmatrix}\end{split}\]EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[] sage: p = a0*x^4 + 4*a1*x^3*y + 6*a2*x^2*y^2 + 4*a3*x*y^3 + a4*y^4 sage: inv = invariant_theory.binary_quartic(p, x, y) sage: g = inv.g_covariant(); g a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3*y - 2*a0*a3*x^3*y + 3*a2^2*x^2*y^2 - 2*a1*a3*x^2*y^2 - a0*a4*x^2*y^2 + 2*a2*a3*x*y^3 - 2*a1*a4*x*y^3 + a3^2*y^4 - a2*a4*y^4 sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x) sage: inv_inhomogeneous.g_covariant() a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3 - 2*a0*a3*x^3 + 3*a2^2*x^2 - 2*a1*a3*x^2 - a0*a4*x^2 + 2*a2*a3*x - 2*a1*a4*x + a3^2 - a2*a4 sage: g == 1/144 * (p.derivative(x,y)^2 - p.derivative(x,x)*p.derivative(y,y)) True
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x, y']; (a0, a1, a2, a3, a4, x, y,) = R._first_ngens(7) >>> p = a0*x**Integer(4) + Integer(4)*a1*x**Integer(3)*y + Integer(6)*a2*x**Integer(2)*y**Integer(2) + Integer(4)*a3*x*y**Integer(3) + a4*y**Integer(4) >>> inv = invariant_theory.binary_quartic(p, x, y) >>> g = inv.g_covariant(); g a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3*y - 2*a0*a3*x^3*y + 3*a2^2*x^2*y^2 - 2*a1*a3*x^2*y^2 - a0*a4*x^2*y^2 + 2*a2*a3*x*y^3 - 2*a1*a4*x*y^3 + a3^2*y^4 - a2*a4*y^4 >>> inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=Integer(1)), x) >>> inv_inhomogeneous.g_covariant() a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3 - 2*a0*a3*x^3 + 3*a2^2*x^2 - 2*a1*a3*x^2 - a0*a4*x^2 + 2*a2*a3*x - 2*a1*a4*x + a3^2 - a2*a4 >>> g == Integer(1)/Integer(144) * (p.derivative(x,y)**Integer(2) - p.derivative(x,x)*p.derivative(y,y)) True
R.<a0, a1, a2, a3, a4, x, y> = QQ[] p = a0*x^4 + 4*a1*x^3*y + 6*a2*x^2*y^2 + 4*a3*x*y^3 + a4*y^4 inv = invariant_theory.binary_quartic(p, x, y) g = inv.g_covariant(); g inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x) inv_inhomogeneous.g_covariant() g == 1/144 * (p.derivative(x,y)^2 - p.derivative(x,x)*p.derivative(y,y))
- h_covariant()[source]¶
The h-covariant of a binary quartic.
OUTPUT: the h-covariant of the quartic
\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = \frac{1}{144} \begin{pmatrix} \frac{\partial^2 f}{\partial x \partial x} \end{pmatrix}\end{split}\]EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[] sage: p = a0*x^4 + 4*a1*x^3*y + 6*a2*x^2*y^2 + 4*a3*x*y^3 + a4*y^4 sage: inv = invariant_theory.binary_quartic(p, x, y) sage: h = inv.h_covariant(); h -2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5*y + 9*a0*a2^2*x^5*y - 2*a0*a1*a3*x^5*y - a0^2*a4*x^5*y - 10*a1^2*a3*x^4*y^2 + 15*a0*a2*a3*x^4*y^2 - 5*a0*a1*a4*x^4*y^2 + 10*a0*a3^2*x^3*y^3 - 10*a1^2*a4*x^3*y^3 + 10*a1*a3^2*x^2*y^4 - 15*a1*a2*a4*x^2*y^4 + 5*a0*a3*a4*x^2*y^4 + 6*a2*a3^2*x*y^5 - 9*a2^2*a4*x*y^5 + 2*a1*a3*a4*x*y^5 + a0*a4^2*x*y^5 + 2*a3^3*y^6 - 3*a2*a3*a4*y^6 + a1*a4^2*y^6 sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x) sage: inv_inhomogeneous.h_covariant() -2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5 + 9*a0*a2^2*x^5 - 2*a0*a1*a3*x^5 - a0^2*a4*x^5 - 10*a1^2*a3*x^4 + 15*a0*a2*a3*x^4 - 5*a0*a1*a4*x^4 + 10*a0*a3^2*x^3 - 10*a1^2*a4*x^3 + 10*a1*a3^2*x^2 - 15*a1*a2*a4*x^2 + 5*a0*a3*a4*x^2 + 6*a2*a3^2*x - 9*a2^2*a4*x + 2*a1*a3*a4*x + a0*a4^2*x + 2*a3^3 - 3*a2*a3*a4 + a1*a4^2 sage: g = inv.g_covariant() sage: h == 1/8 * (p.derivative(x)*g.derivative(y) - p.derivative(y)*g.derivative(x)) True
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x, y']; (a0, a1, a2, a3, a4, x, y,) = R._first_ngens(7) >>> p = a0*x**Integer(4) + Integer(4)*a1*x**Integer(3)*y + Integer(6)*a2*x**Integer(2)*y**Integer(2) + Integer(4)*a3*x*y**Integer(3) + a4*y**Integer(4) >>> inv = invariant_theory.binary_quartic(p, x, y) >>> h = inv.h_covariant(); h -2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5*y + 9*a0*a2^2*x^5*y - 2*a0*a1*a3*x^5*y - a0^2*a4*x^5*y - 10*a1^2*a3*x^4*y^2 + 15*a0*a2*a3*x^4*y^2 - 5*a0*a1*a4*x^4*y^2 + 10*a0*a3^2*x^3*y^3 - 10*a1^2*a4*x^3*y^3 + 10*a1*a3^2*x^2*y^4 - 15*a1*a2*a4*x^2*y^4 + 5*a0*a3*a4*x^2*y^4 + 6*a2*a3^2*x*y^5 - 9*a2^2*a4*x*y^5 + 2*a1*a3*a4*x*y^5 + a0*a4^2*x*y^5 + 2*a3^3*y^6 - 3*a2*a3*a4*y^6 + a1*a4^2*y^6 >>> inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=Integer(1)), x) >>> inv_inhomogeneous.h_covariant() -2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5 + 9*a0*a2^2*x^5 - 2*a0*a1*a3*x^5 - a0^2*a4*x^5 - 10*a1^2*a3*x^4 + 15*a0*a2*a3*x^4 - 5*a0*a1*a4*x^4 + 10*a0*a3^2*x^3 - 10*a1^2*a4*x^3 + 10*a1*a3^2*x^2 - 15*a1*a2*a4*x^2 + 5*a0*a3*a4*x^2 + 6*a2*a3^2*x - 9*a2^2*a4*x + 2*a1*a3*a4*x + a0*a4^2*x + 2*a3^3 - 3*a2*a3*a4 + a1*a4^2 >>> g = inv.g_covariant() >>> h == Integer(1)/Integer(8) * (p.derivative(x)*g.derivative(y) - p.derivative(y)*g.derivative(x)) True
R.<a0, a1, a2, a3, a4, x, y> = QQ[] p = a0*x^4 + 4*a1*x^3*y + 6*a2*x^2*y^2 + 4*a3*x*y^3 + a4*y^4 inv = invariant_theory.binary_quartic(p, x, y) h = inv.h_covariant(); h inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x) inv_inhomogeneous.h_covariant() g = inv.g_covariant() h == 1/8 * (p.derivative(x)*g.derivative(y) - p.derivative(y)*g.derivative(x))
- monomials()[source]¶
List the basis monomials in the form.
OUTPUT:
A tuple of monomials. They are in the same order as
coeffs().EXAMPLES:
sage: R.<x,y> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4) sage: quartic.monomials() (y^4, x*y^3, x^2*y^2, x^3*y, x^4)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4)) >>> quartic.monomials() (y^4, x*y^3, x^2*y^2, x^3*y, x^4)
R.<x,y> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4) quartic.monomials()
- scaled_coeffs()[source]¶
The coefficients of a binary quartic.
Given
\[f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4\]this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] sage: quartic = a0*x1^4 + 4*a1*x1^3*x0 + 6*a2*x1^2*x0^2 + 4*a3*x1*x0^3 + a4*x0^4 sage: inv = invariant_theory.binary_quartic(quartic, x0, x1) sage: inv.scaled_coeffs() (a0, a1, a2, a3, a4) sage: R.<a0, a1, a2, a3, a4, x> = QQ[] sage: quartic = a0 + 4*a1*x + 6*a2*x^2 + 4*a3*x^3 + a4*x^4 sage: inv = invariant_theory.binary_quartic(quartic, x) sage: inv.scaled_coeffs() (a0, a1, a2, a3, a4)
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, x0, x1']; (a0, a1, a2, a3, a4, x0, x1,) = R._first_ngens(7) >>> quartic = a0*x1**Integer(4) + Integer(4)*a1*x1**Integer(3)*x0 + Integer(6)*a2*x1**Integer(2)*x0**Integer(2) + Integer(4)*a3*x1*x0**Integer(3) + a4*x0**Integer(4) >>> inv = invariant_theory.binary_quartic(quartic, x0, x1) >>> inv.scaled_coeffs() (a0, a1, a2, a3, a4) >>> R = QQ['a0, a1, a2, a3, a4, x']; (a0, a1, a2, a3, a4, x,) = R._first_ngens(6) >>> quartic = a0 + Integer(4)*a1*x + Integer(6)*a2*x**Integer(2) + Integer(4)*a3*x**Integer(3) + a4*x**Integer(4) >>> inv = invariant_theory.binary_quartic(quartic, x) >>> inv.scaled_coeffs() (a0, a1, a2, a3, a4)
R.<a0, a1, a2, a3, a4, x0, x1> = QQ[] quartic = a0*x1^4 + 4*a1*x1^3*x0 + 6*a2*x1^2*x0^2 + 4*a3*x1*x0^3 + a4*x0^4 inv = invariant_theory.binary_quartic(quartic, x0, x1) inv.scaled_coeffs() R.<a0, a1, a2, a3, a4, x> = QQ[] quartic = a0 + 4*a1*x + 6*a2*x^2 + 4*a3*x^3 + a4*x^4 inv = invariant_theory.binary_quartic(quartic, x) inv.scaled_coeffs()
- class sage.rings.invariants.invariant_theory.BinaryQuintic(n, d, polynomial, *args)[source]¶
Bases:
AlgebraicFormInvariant theory of a binary quintic form.
You should use the
invariant_theoryfactory object to construct instances of this class. Seebinary_quintic()for details.REFERENCES:
For a description of all invariants and covariants of a binary quintic, see section 73 of [Cle1872].
- A_invariant()[source]¶
Return the invariant \(A\) of a binary quintic.
OUTPUT: the \(A\)-invariant of the binary quintic
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.A_invariant() 4/625*a2^2*a3^2 - 12/625*a1*a3^3 - 12/625*a2^3*a4 + 38/625*a1*a2*a3*a4 + 6/125*a0*a3^2*a4 - 18/625*a1^2*a4^2 - 16/125*a0*a2*a4^2 + 6/125*a1*a2^2*a5 - 16/125*a1^2*a3*a5 - 2/25*a0*a2*a3*a5 + 4/5*a0*a1*a4*a5 - 2*a0^2*a5^2
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.A_invariant() 4/625*a2^2*a3^2 - 12/625*a1*a3^3 - 12/625*a2^3*a4 + 38/625*a1*a2*a3*a4 + 6/125*a0*a3^2*a4 - 18/625*a1^2*a4^2 - 16/125*a0*a2*a4^2 + 6/125*a1*a2^2*a5 - 16/125*a1^2*a3*a5 - 2/25*a0*a2*a3*a5 + 4/5*a0*a1*a4*a5 - 2*a0^2*a5^2
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.A_invariant()
- B_invariant()[source]¶
Return the invariant \(B\) of a binary quintic.
OUTPUT: the \(B\)-invariant of the binary quintic
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.B_invariant() 1/1562500*a2^4*a3^4 - 3/781250*a1*a2^2*a3^5 + 9/1562500*a1^2*a3^6 - 3/781250*a2^5*a3^2*a4 + 37/1562500*a1*a2^3*a3^3*a4 - 57/1562500*a1^2*a2*a3^4*a4 + 3/312500*a0*a2^2*a3^4*a4 ... + 8/625*a0^2*a1^2*a4^2*a5^2 - 4/125*a0^3*a2*a4^2*a5^2 - 16/3125*a1^5*a5^3 + 4/125*a0*a1^3*a2*a5^3 - 6/125*a0^2*a1*a2^2*a5^3 - 4/125*a0^2*a1^2*a3*a5^3 + 2/25*a0^3*a2*a3*a5^3
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.B_invariant() 1/1562500*a2^4*a3^4 - 3/781250*a1*a2^2*a3^5 + 9/1562500*a1^2*a3^6 - 3/781250*a2^5*a3^2*a4 + 37/1562500*a1*a2^3*a3^3*a4 - 57/1562500*a1^2*a2*a3^4*a4 + 3/312500*a0*a2^2*a3^4*a4 ... + 8/625*a0^2*a1^2*a4^2*a5^2 - 4/125*a0^3*a2*a4^2*a5^2 - 16/3125*a1^5*a5^3 + 4/125*a0*a1^3*a2*a5^3 - 6/125*a0^2*a1*a2^2*a5^3 - 4/125*a0^2*a1^2*a3*a5^3 + 2/25*a0^3*a2*a3*a5^3
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.B_invariant()
- C_invariant()[source]¶
Return the invariant \(C\) of a binary quintic.
OUTPUT: the \(C\)-invariant of the binary quintic
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.C_invariant() -3/1953125000*a2^6*a3^6 + 27/1953125000*a1*a2^4*a3^7 - 249/7812500000*a1^2*a2^2*a3^8 - 3/78125000*a0*a2^3*a3^8 + 3/976562500*a1^3*a3^9 + 27/156250000*a0*a1*a2*a3^9 ... + 192/15625*a0^2*a1^3*a2^2*a3*a5^4 - 36/3125*a0^3*a1*a2^3*a3*a5^4 + 24/15625*a0^2*a1^4*a3^2*a5^4 - 24/3125*a0^3*a1^2*a2*a3^2*a5^4 + 6/625*a0^4*a2^2*a3^2*a5^4
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.C_invariant() -3/1953125000*a2^6*a3^6 + 27/1953125000*a1*a2^4*a3^7 - 249/7812500000*a1^2*a2^2*a3^8 - 3/78125000*a0*a2^3*a3^8 + 3/976562500*a1^3*a3^9 + 27/156250000*a0*a1*a2*a3^9 ... + 192/15625*a0^2*a1^3*a2^2*a3*a5^4 - 36/3125*a0^3*a1*a2^3*a3*a5^4 + 24/15625*a0^2*a1^4*a3^2*a5^4 - 24/3125*a0^3*a1^2*a2*a3^2*a5^4 + 6/625*a0^4*a2^2*a3^2*a5^4
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.C_invariant()
- H_covariant(as_form=False)[source]¶
Return the covariant \(H\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT: the \(H\)-covariant of the binary quintic as polynomial or as binary form
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.H_covariant() -2/25*a4^2*x0^6 + 1/5*a3*a5*x0^6 - 3/25*a3*a4*x0^5*x1 + 3/5*a2*a5*x0^5*x1 - 3/25*a3^2*x0^4*x1^2 + 3/25*a2*a4*x0^4*x1^2 + 6/5*a1*a5*x0^4*x1^2 - 4/25*a2*a3*x0^3*x1^3 + 14/25*a1*a4*x0^3*x1^3 + 2*a0*a5*x0^3*x1^3 - 3/25*a2^2*x0^2*x1^4 + 3/25*a1*a3*x0^2*x1^4 + 6/5*a0*a4*x0^2*x1^4 - 3/25*a1*a2*x0*x1^5 + 3/5*a0*a3*x0*x1^5 - 2/25*a1^2*x1^6 + 1/5*a0*a2*x1^6 sage: quintic.H_covariant(as_form=True) Binary sextic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.H_covariant() -2/25*a4^2*x0^6 + 1/5*a3*a5*x0^6 - 3/25*a3*a4*x0^5*x1 + 3/5*a2*a5*x0^5*x1 - 3/25*a3^2*x0^4*x1^2 + 3/25*a2*a4*x0^4*x1^2 + 6/5*a1*a5*x0^4*x1^2 - 4/25*a2*a3*x0^3*x1^3 + 14/25*a1*a4*x0^3*x1^3 + 2*a0*a5*x0^3*x1^3 - 3/25*a2^2*x0^2*x1^4 + 3/25*a1*a3*x0^2*x1^4 + 6/5*a0*a4*x0^2*x1^4 - 3/25*a1*a2*x0*x1^5 + 3/5*a0*a3*x0*x1^5 - 2/25*a1^2*x1^6 + 1/5*a0*a2*x1^6 >>> quintic.H_covariant(as_form=True) Binary sextic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.H_covariant() quintic.H_covariant(as_form=True)
- R_invariant()[source]¶
Return the invariant \(R\) of a binary quintic.
OUTPUT: the \(R\)-invariant of the binary quintic
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.R_invariant() 3/3906250000000*a1^2*a2^5*a3^11 - 3/976562500000*a0*a2^6*a3^11 - 51/7812500000000*a1^3*a2^3*a3^12 + 27/976562500000*a0*a1*a2^4*a3^12 + 27/1953125000000*a1^4*a2*a3^13 - 81/1562500000000*a0*a1^2*a2^2*a3^13 ... + 384/9765625*a0*a1^10*a5^7 - 192/390625*a0^2*a1^8*a2*a5^7 + 192/78125*a0^3*a1^6*a2^2*a5^7 - 96/15625*a0^4*a1^4*a2^3*a5^7 + 24/3125*a0^5*a1^2*a2^4*a5^7 - 12/3125*a0^6*a2^5*a5^7
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.R_invariant() 3/3906250000000*a1^2*a2^5*a3^11 - 3/976562500000*a0*a2^6*a3^11 - 51/7812500000000*a1^3*a2^3*a3^12 + 27/976562500000*a0*a1*a2^4*a3^12 + 27/1953125000000*a1^4*a2*a3^13 - 81/1562500000000*a0*a1^2*a2^2*a3^13 ... + 384/9765625*a0*a1^10*a5^7 - 192/390625*a0^2*a1^8*a2*a5^7 + 192/78125*a0^3*a1^6*a2^2*a5^7 - 96/15625*a0^4*a1^4*a2^3*a5^7 + 24/3125*a0^5*a1^2*a2^4*a5^7 - 12/3125*a0^6*a2^5*a5^7
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.R_invariant()
- T_covariant(as_form=False)[source]¶
Return the covariant \(T\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT: the \(T\)-covariant of the binary quintic as polynomial or as binary form
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.T_covariant() 2/125*a4^3*x0^9 - 3/50*a3*a4*a5*x0^9 + 1/10*a2*a5^2*x0^9 + 9/250*a3*a4^2*x0^8*x1 - 3/25*a3^2*a5*x0^8*x1 + 1/50*a2*a4*a5*x0^8*x1 + 2/5*a1*a5^2*x0^8*x1 + 3/250*a3^2*a4*x0^7*x1^2 + 8/125*a2*a4^2*x0^7*x1^2 ... 11/25*a0*a1*a4*x0^2*x1^7 - a0^2*a5*x0^2*x1^7 - 9/250*a1^2*a2*x0*x1^8 + 3/25*a0*a2^2*x0*x1^8 - 1/50*a0*a1*a3*x0*x1^8 - 2/5*a0^2*a4*x0*x1^8 - 2/125*a1^3*x1^9 + 3/50*a0*a1*a2*x1^9 - 1/10*a0^2*a3*x1^9 sage: quintic.T_covariant(as_form=True) Binary nonic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.T_covariant() 2/125*a4^3*x0^9 - 3/50*a3*a4*a5*x0^9 + 1/10*a2*a5^2*x0^9 + 9/250*a3*a4^2*x0^8*x1 - 3/25*a3^2*a5*x0^8*x1 + 1/50*a2*a4*a5*x0^8*x1 + 2/5*a1*a5^2*x0^8*x1 + 3/250*a3^2*a4*x0^7*x1^2 + 8/125*a2*a4^2*x0^7*x1^2 ... 11/25*a0*a1*a4*x0^2*x1^7 - a0^2*a5*x0^2*x1^7 - 9/250*a1^2*a2*x0*x1^8 + 3/25*a0*a2^2*x0*x1^8 - 1/50*a0*a1*a3*x0*x1^8 - 2/5*a0^2*a4*x0*x1^8 - 2/125*a1^3*x1^9 + 3/50*a0*a1*a2*x1^9 - 1/10*a0^2*a3*x1^9 >>> quintic.T_covariant(as_form=True) Binary nonic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.T_covariant() quintic.T_covariant(as_form=True)
- alpha_covariant(as_form=False)[source]¶
Return the covariant \(\alpha\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\alpha\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.alpha_covariant() 1/2500*a2^2*a3^3*x0 - 3/2500*a1*a3^4*x0 - 1/625*a2^3*a3*a4*x0 + 3/625*a1*a2*a3^2*a4*x0 + 3/625*a0*a3^3*a4*x0 + 2/625*a1*a2^2*a4^2*x0 - 6/625*a1^2*a3*a4^2*x0 - 12/625*a0*a2*a3*a4^2*x0 + 24/625*a0*a1*a4^3*x0 ... - 12/625*a1^2*a2*a3*a5*x1 - 1/125*a0*a2^2*a3*a5*x1 + 8/125*a0*a1*a3^2*a5*x1 + 24/625*a1^3*a4*a5*x1 - 8/125*a0*a1*a2*a4*a5*x1 - 4/25*a0^2*a3*a4*a5*x1 - 4/25*a0*a1^2*a5^2*x1 + 2/5*a0^2*a2*a5^2*x1 sage: quintic.alpha_covariant(as_form=True) Binary monic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.alpha_covariant() 1/2500*a2^2*a3^3*x0 - 3/2500*a1*a3^4*x0 - 1/625*a2^3*a3*a4*x0 + 3/625*a1*a2*a3^2*a4*x0 + 3/625*a0*a3^3*a4*x0 + 2/625*a1*a2^2*a4^2*x0 - 6/625*a1^2*a3*a4^2*x0 - 12/625*a0*a2*a3*a4^2*x0 + 24/625*a0*a1*a4^3*x0 ... - 12/625*a1^2*a2*a3*a5*x1 - 1/125*a0*a2^2*a3*a5*x1 + 8/125*a0*a1*a3^2*a5*x1 + 24/625*a1^3*a4*a5*x1 - 8/125*a0*a1*a2*a4*a5*x1 - 4/25*a0^2*a3*a4*a5*x1 - 4/25*a0*a1^2*a5^2*x1 + 2/5*a0^2*a2*a5^2*x1 >>> quintic.alpha_covariant(as_form=True) Binary monic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.alpha_covariant() quintic.alpha_covariant(as_form=True)
- arithmetic_invariants()[source]¶
Return a set of generating arithmetic invariants of a binary quintic.
An arithmetic invariants is an invariant whose coefficients are integers for a general binary quintic. They are linear combinations of the Clebsch invariants, such that they still generate the ring of invariants.
OUTPUT: the arithmetic invariants of the binary quintic. They are given by
\[\begin{split}\begin{aligned} I_4 & = 2^{-1} \cdot 5^4 \cdot A \\ I_8 & = 5^5 \cdot (2^{-1} \cdot 47 \cdot A^2 - 2^2 \cdot B) \\ I_{12} & = 5^{10} \cdot (2^{-1} \cdot 3 \cdot A^3 - 2^5 \cdot 3^{-1} \cdot C) \\ I_{18} & = 2^8 \cdot 3^{-1} \cdot 5^{15} \cdot R \\ \end{aligned}\end{split}\]where \(A\), \(B\), \(C\) and \(R\) are the
BinaryQuintic.clebsch_invariants().EXAMPLES:
sage: R.<x0, x1> = QQ[] sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.arithmetic_invariants() {'I12': -1156502613073152, 'I18': -12712872348048797642752, 'I4': -138016, 'I8': 14164936192}
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(2)*x1**Integer(5) + Integer(4)*x1**Integer(4)*x0 + Integer(5)*x1**Integer(3)*x0**Integer(2) + Integer(7)*x1**Integer(2)*x0**Integer(3) - Integer(11)*x1*x0**Integer(4) + x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.arithmetic_invariants() {'I12': -1156502613073152, 'I18': -12712872348048797642752, 'I4': -138016, 'I8': 14164936192}
R.<x0, x1> = QQ[] p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.arithmetic_invariants()
We can check that the coefficients of the invariants have no common divisor for a general quintic form:
sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: invs = quintic.arithmetic_invariants() sage: [invs[x].content() for x in invs] [1, 1, 1, 1]
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> invs = quintic.arithmetic_invariants() >>> [invs[x].content() for x in invs] [1, 1, 1, 1]
R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) invs = quintic.arithmetic_invariants() [invs[x].content() for x in invs]
- beta_covariant(as_form=False)[source]¶
Return the covariant \(\beta\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\beta\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.beta_covariant() -1/62500*a2^3*a3^4*x0 + 9/125000*a1*a2*a3^5*x0 - 27/125000*a0*a3^6*x0 + 13/125000*a2^4*a3^2*a4*x0 - 31/62500*a1*a2^2*a3^3*a4*x0 - 3/62500*a1^2*a3^4*a4*x0 + 27/15625*a0*a2*a3^4*a4*x0 ... - 16/125*a0^2*a1*a3^2*a5^2*x1 - 28/625*a0*a1^3*a4*a5^2*x1 + 6/125*a0^2*a1*a2*a4*a5^2*x1 + 8/25*a0^3*a3*a4*a5^2*x1 + 4/25*a0^2*a1^2*a5^3*x1 - 2/5*a0^3*a2*a5^3*x1 sage: quintic.beta_covariant(as_form=True) Binary monic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.beta_covariant() -1/62500*a2^3*a3^4*x0 + 9/125000*a1*a2*a3^5*x0 - 27/125000*a0*a3^6*x0 + 13/125000*a2^4*a3^2*a4*x0 - 31/62500*a1*a2^2*a3^3*a4*x0 - 3/62500*a1^2*a3^4*a4*x0 + 27/15625*a0*a2*a3^4*a4*x0 ... - 16/125*a0^2*a1*a3^2*a5^2*x1 - 28/625*a0*a1^3*a4*a5^2*x1 + 6/125*a0^2*a1*a2*a4*a5^2*x1 + 8/25*a0^3*a3*a4*a5^2*x1 + 4/25*a0^2*a1^2*a5^3*x1 - 2/5*a0^3*a2*a5^3*x1 >>> quintic.beta_covariant(as_form=True) Binary monic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.beta_covariant() quintic.beta_covariant(as_form=True)
- canonical_form(reduce_gcd=False)[source]¶
Return a canonical representative of the quintic.
Given a binary quintic \(f\) with coefficients in a field \(K\), returns a canonical representative of the \(GL(2,\bar{K})\)-orbit of the quintic, where \(\bar{K}\) is an algebraic closure of \(K\). This means that two binary quintics \(f\) and \(g\) are \(GL(2,\bar{K})\)-equivalent if and only if their canonical forms are the same.
INPUT:
reduce_gcd– if set toTrue, then a variant of this canonical form is computed where the coefficients are coprime integers. The obtained form is then unique up to multiplication by a unit. See alsobinary_quintic_from_invariants()’.
OUTPUT:
A canonical \(GL(2,\bar{K})\)-equivalent binary quintic.
EXAMPLES:
sage: R.<x0, x1> = QQ[] sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 sage: f = invariant_theory.binary_quintic(p, x0, x1) sage: g = matrix(QQ, [[11,5],[7,2]]) sage: gf = f.transformed(g) sage: f.canonical_form() == gf.canonical_form() True sage: h = f.canonical_form(reduce_gcd=True) sage: gcd(h.coeffs()) 1
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(2)*x1**Integer(5) + Integer(4)*x1**Integer(4)*x0 + Integer(5)*x1**Integer(3)*x0**Integer(2) + Integer(7)*x1**Integer(2)*x0**Integer(3) - Integer(11)*x1*x0**Integer(4) + x0**Integer(5) >>> f = invariant_theory.binary_quintic(p, x0, x1) >>> g = matrix(QQ, [[Integer(11),Integer(5)],[Integer(7),Integer(2)]]) >>> gf = f.transformed(g) >>> f.canonical_form() == gf.canonical_form() True >>> h = f.canonical_form(reduce_gcd=True) >>> gcd(h.coeffs()) 1
R.<x0, x1> = QQ[] p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 f = invariant_theory.binary_quintic(p, x0, x1) g = matrix(QQ, [[11,5],[7,2]]) gf = f.transformed(g) f.canonical_form() == gf.canonical_form() h = f.canonical_form(reduce_gcd=True) gcd(h.coeffs())
- clebsch_invariants(as_tuple=False)[source]¶
Return the invariants of a binary quintic as described by Clebsch.
The following invariants are returned: \(A\), \(B\), \(C\) and \(R\).
OUTPUT: the Clebsch invariants of the binary quintic
EXAMPLES:
sage: R.<x0, x1> = QQ[] sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.clebsch_invariants() {'A': -276032/625, 'B': 4983526016/390625, 'C': -247056495846408/244140625, 'R': -148978972828696847376/30517578125} sage: quintic.clebsch_invariants(as_tuple=True) (-276032/625, 4983526016/390625, -247056495846408/244140625, -148978972828696847376/30517578125)
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(2)*x1**Integer(5) + Integer(4)*x1**Integer(4)*x0 + Integer(5)*x1**Integer(3)*x0**Integer(2) + Integer(7)*x1**Integer(2)*x0**Integer(3) - Integer(11)*x1*x0**Integer(4) + x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.clebsch_invariants() {'A': -276032/625, 'B': 4983526016/390625, 'C': -247056495846408/244140625, 'R': -148978972828696847376/30517578125} >>> quintic.clebsch_invariants(as_tuple=True) (-276032/625, 4983526016/390625, -247056495846408/244140625, -148978972828696847376/30517578125)
R.<x0, x1> = QQ[] p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.clebsch_invariants() quintic.clebsch_invariants(as_tuple=True)
- coeffs()[source]¶
The coefficients of a binary quintic.
Given
\[f(x) = a_0 x_1^5 + a_1 x_0 x_1^4 + a_2 x_0^2 x_1^3 + a_3 x_0^3 x_1^2 + a_4 x_0^4 x_1 + a_5 x_1^5\]this function returns \(a = (a_0, a_1, a_2, a_3, a_4, a_5)\)
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.coeffs() (a0, a1, a2, a3, a4, a5) sage: R.<a0, a1, a2, a3, a4, a5, x> = QQ[] sage: p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5 sage: quintic = invariant_theory.binary_quintic(p, x) sage: quintic.coeffs() (a0, a1, a2, a3, a4, a5)
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.coeffs() (a0, a1, a2, a3, a4, a5) >>> R = QQ['a0, a1, a2, a3, a4, a5, x']; (a0, a1, a2, a3, a4, a5, x,) = R._first_ngens(7) >>> p = a0 + a1*x + a2*x**Integer(2) + a3*x**Integer(3) + a4*x**Integer(4) + a5*x**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x) >>> quintic.coeffs() (a0, a1, a2, a3, a4, a5)
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.coeffs() R.<a0, a1, a2, a3, a4, a5, x> = QQ[] p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5 quintic = invariant_theory.binary_quintic(p, x) quintic.coeffs()
- delta_covariant(as_form=False)[source]¶
Return the covariant \(\delta\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\delta\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.delta_covariant() 1/1562500000*a2^6*a3^7*x0 - 9/1562500000*a1*a2^4*a3^8*x0 + 9/625000000*a1^2*a2^2*a3^9*x0 + 9/781250000*a0*a2^3*a3^9*x0 - 9/1562500000*a1^3*a3^10*x0 - 81/1562500000*a0*a1*a2*a3^10*x0 ... + 64/3125*a0^3*a1^3*a2^2*a5^5*x1 - 12/625*a0^4*a1*a2^3*a5^5*x1 + 16/3125*a0^3*a1^4*a3*a5^5*x1 - 16/625*a0^4*a1^2*a2*a3*a5^5*x1 + 4/125*a0^5*a2^2*a3*a5^5*x1 sage: quintic.delta_covariant(as_form=True) Binary monic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.delta_covariant() 1/1562500000*a2^6*a3^7*x0 - 9/1562500000*a1*a2^4*a3^8*x0 + 9/625000000*a1^2*a2^2*a3^9*x0 + 9/781250000*a0*a2^3*a3^9*x0 - 9/1562500000*a1^3*a3^10*x0 - 81/1562500000*a0*a1*a2*a3^10*x0 ... + 64/3125*a0^3*a1^3*a2^2*a5^5*x1 - 12/625*a0^4*a1*a2^3*a5^5*x1 + 16/3125*a0^3*a1^4*a3*a5^5*x1 - 16/625*a0^4*a1^2*a2*a3*a5^5*x1 + 4/125*a0^5*a2^2*a3*a5^5*x1 >>> quintic.delta_covariant(as_form=True) Binary monic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.delta_covariant() quintic.delta_covariant(as_form=True)
- classmethod from_invariants(invariants, x, z, *args, **kwargs)[source]¶
Construct a binary quintic from its invariants.
This function constructs a binary quintic whose invariants equal the ones provided as argument up to scaling.
INPUT:
invariants– list or tuple of invariants that are used to reconstruct the binary quintic
OUTPUT: a BinaryQuintic
EXAMPLES:
sage: R.<x,y> = QQ[] sage: from sage.rings.invariants.invariant_theory import BinaryQuintic sage: BinaryQuintic.from_invariants([3,6,12], x, y) Binary quintic with coefficients (0, 1, 0, 0, 1, 0)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> from sage.rings.invariants.invariant_theory import BinaryQuintic >>> BinaryQuintic.from_invariants([Integer(3),Integer(6),Integer(12)], x, y) Binary quintic with coefficients (0, 1, 0, 0, 1, 0)
R.<x,y> = QQ[] from sage.rings.invariants.invariant_theory import BinaryQuintic BinaryQuintic.from_invariants([3,6,12], x, y)
- gamma_covariant(as_form=False)[source]¶
Return the covariant \(\gamma\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\gamma\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.gamma_covariant() 1/156250000*a2^5*a3^6*x0 - 3/62500000*a1*a2^3*a3^7*x0 + 27/312500000*a1^2*a2*a3^8*x0 + 27/312500000*a0*a2^2*a3^8*x0 - 81/312500000*a0*a1*a3^9*x0 - 19/312500000*a2^6*a3^4*a4*x0 ... - 32/3125*a0^2*a1^3*a2^2*a5^4*x1 + 6/625*a0^3*a1*a2^3*a5^4*x1 - 8/3125*a0^2*a1^4*a3*a5^4*x1 + 8/625*a0^3*a1^2*a2*a3*a5^4*x1 - 2/125*a0^4*a2^2*a3*a5^4*x1 sage: quintic.gamma_covariant(as_form=True) Binary monic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.gamma_covariant() 1/156250000*a2^5*a3^6*x0 - 3/62500000*a1*a2^3*a3^7*x0 + 27/312500000*a1^2*a2*a3^8*x0 + 27/312500000*a0*a2^2*a3^8*x0 - 81/312500000*a0*a1*a3^9*x0 - 19/312500000*a2^6*a3^4*a4*x0 ... - 32/3125*a0^2*a1^3*a2^2*a5^4*x1 + 6/625*a0^3*a1*a2^3*a5^4*x1 - 8/3125*a0^2*a1^4*a3*a5^4*x1 + 8/625*a0^3*a1^2*a2*a3*a5^4*x1 - 2/125*a0^4*a2^2*a3*a5^4*x1 >>> quintic.gamma_covariant(as_form=True) Binary monic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.gamma_covariant() quintic.gamma_covariant(as_form=True)
- i_covariant(as_form=False)[source]¶
Return the covariant \(i\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT: the \(i\)-covariant of the binary quintic as polynomial or as binary form
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.i_covariant() 3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2 + 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1 - 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1 + 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2 + 2/5*a0*a4*x1^2 sage: quintic.i_covariant(as_form=True) Binary quadratic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.i_covariant() 3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2 + 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1 - 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1 + 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2 + 2/5*a0*a4*x1^2 >>> quintic.i_covariant(as_form=True) Binary quadratic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.i_covariant() quintic.i_covariant(as_form=True)
- invariants(type='clebsch')[source]¶
Return a tuple of invariants of a binary quintic.
INPUT:
type– the type of invariants to return; the default choice is to return the Clebsch invariants
OUTPUT: the invariants of the binary quintic
EXAMPLES:
sage: R.<x0, x1> = QQ[] sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.invariants() (-276032/625, 4983526016/390625, -247056495846408/244140625, -148978972828696847376/30517578125) sage: quintic.invariants('unknown') Traceback (most recent call last): ... ValueError: unknown type of invariants unknown for a binary quintic
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(2)*x1**Integer(5) + Integer(4)*x1**Integer(4)*x0 + Integer(5)*x1**Integer(3)*x0**Integer(2) + Integer(7)*x1**Integer(2)*x0**Integer(3) - Integer(11)*x1*x0**Integer(4) + x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.invariants() (-276032/625, 4983526016/390625, -247056495846408/244140625, -148978972828696847376/30517578125) >>> quintic.invariants('unknown') Traceback (most recent call last): ... ValueError: unknown type of invariants unknown for a binary quintic
R.<x0, x1> = QQ[] p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.invariants() quintic.invariants('unknown')
- j_covariant(as_form=False)[source]¶
Return the covariant \(j\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT: the \(j\)-covariant of the binary quintic as polynomial or as binary form
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.j_covariant() -3/500*a3^3*x0^3 + 3/125*a2*a3*a4*x0^3 - 6/125*a1*a4^2*x0^3 - 3/50*a2^2*a5*x0^3 + 3/25*a1*a3*a5*x0^3 - 3/500*a2*a3^2*x0^2*x1 + 3/250*a2^2*a4*x0^2*x1 + 3/125*a1*a3*a4*x0^2*x1 - 6/25*a0*a4^2*x0^2*x1 - 3/25*a1*a2*a5*x0^2*x1 + 3/5*a0*a3*a5*x0^2*x1 - 3/500*a2^2*a3*x0*x1^2 + 3/250*a1*a3^2*x0*x1^2 + 3/125*a1*a2*a4*x0*x1^2 - 3/25*a0*a3*a4*x0*x1^2 - 6/25*a1^2*a5*x0*x1^2 + 3/5*a0*a2*a5*x0*x1^2 - 3/500*a2^3*x1^3 + 3/125*a1*a2*a3*x1^3 - 3/50*a0*a3^2*x1^3 - 6/125*a1^2*a4*x1^3 + 3/25*a0*a2*a4*x1^3 sage: quintic.j_covariant(as_form=True) Binary cubic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.j_covariant() -3/500*a3^3*x0^3 + 3/125*a2*a3*a4*x0^3 - 6/125*a1*a4^2*x0^3 - 3/50*a2^2*a5*x0^3 + 3/25*a1*a3*a5*x0^3 - 3/500*a2*a3^2*x0^2*x1 + 3/250*a2^2*a4*x0^2*x1 + 3/125*a1*a3*a4*x0^2*x1 - 6/25*a0*a4^2*x0^2*x1 - 3/25*a1*a2*a5*x0^2*x1 + 3/5*a0*a3*a5*x0^2*x1 - 3/500*a2^2*a3*x0*x1^2 + 3/250*a1*a3^2*x0*x1^2 + 3/125*a1*a2*a4*x0*x1^2 - 3/25*a0*a3*a4*x0*x1^2 - 6/25*a1^2*a5*x0*x1^2 + 3/5*a0*a2*a5*x0*x1^2 - 3/500*a2^3*x1^3 + 3/125*a1*a2*a3*x1^3 - 3/50*a0*a3^2*x1^3 - 6/125*a1^2*a4*x1^3 + 3/25*a0*a2*a4*x1^3 >>> quintic.j_covariant(as_form=True) Binary cubic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.j_covariant() quintic.j_covariant(as_form=True)
- monomials()[source]¶
List the basis monomials of the form.
This function lists a basis of monomials of the space of binary quintics of which this form is an element.
OUTPUT:
A tuple of monomials. They are in the same order as
coeffs().EXAMPLES:
sage: R.<x,y> = QQ[] sage: quintic = invariant_theory.binary_quintic(x^5 + y^5) sage: quintic.monomials() (y^5, x*y^4, x^2*y^3, x^3*y^2, x^4*y, x^5)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + y**Integer(5)) >>> quintic.monomials() (y^5, x*y^4, x^2*y^3, x^3*y^2, x^4*y, x^5)
R.<x,y> = QQ[] quintic = invariant_theory.binary_quintic(x^5 + y^5) quintic.monomials()
- scaled_coeffs()[source]¶
The coefficients of a binary quintic.
Given
\[f(x) = a_0 x_1^5 + 5 a_1 x_0 x_1^4 + 10 a_2 x_0^2 x_1^3 + 10 a_3 x_0^3 x_1^2 + 5 a_4 x_0^4 x_1 + a_5 x_1^5\]this function returns \(a = (a_0, a_1, a_2, a_3, a_4, a_5)\)
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + 5*a1*x1^4*x0 + 10*a2*x1^3*x0^2 + 10*a3*x1^2*x0^3 + 5*a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.scaled_coeffs() (a0, a1, a2, a3, a4, a5) sage: R.<a0, a1, a2, a3, a4, a5, x> = QQ[] sage: p = a0 + 5*a1*x + 10*a2*x^2 + 10*a3*x^3 + 5*a4*x^4 + a5*x^5 sage: quintic = invariant_theory.binary_quintic(p, x) sage: quintic.scaled_coeffs() (a0, a1, a2, a3, a4, a5)
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + Integer(5)*a1*x1**Integer(4)*x0 + Integer(10)*a2*x1**Integer(3)*x0**Integer(2) + Integer(10)*a3*x1**Integer(2)*x0**Integer(3) + Integer(5)*a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.scaled_coeffs() (a0, a1, a2, a3, a4, a5) >>> R = QQ['a0, a1, a2, a3, a4, a5, x']; (a0, a1, a2, a3, a4, a5, x,) = R._first_ngens(7) >>> p = a0 + Integer(5)*a1*x + Integer(10)*a2*x**Integer(2) + Integer(10)*a3*x**Integer(3) + Integer(5)*a4*x**Integer(4) + a5*x**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x) >>> quintic.scaled_coeffs() (a0, a1, a2, a3, a4, a5)
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + 5*a1*x1^4*x0 + 10*a2*x1^3*x0^2 + 10*a3*x1^2*x0^3 + 5*a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.scaled_coeffs() R.<a0, a1, a2, a3, a4, a5, x> = QQ[] p = a0 + 5*a1*x + 10*a2*x^2 + 10*a3*x^3 + 5*a4*x^4 + a5*x^5 quintic = invariant_theory.binary_quintic(p, x) quintic.scaled_coeffs()
- tau_covariant(as_form=False)[source]¶
Return the covariant \(\tau\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\tau\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.tau_covariant() 1/62500*a2^2*a3^4*x0^2 - 3/62500*a1*a3^5*x0^2 - 1/15625*a2^3*a3^2*a4*x0^2 + 1/6250*a1*a2*a3^3*a4*x0^2 + 3/6250*a0*a3^4*a4*x0^2 - 1/31250*a2^4*a4^2*x0^2 ... - 2/125*a0*a1*a2^2*a4*a5*x1^2 - 4/125*a0*a1^2*a3*a4*a5*x1^2 + 2/25*a0^2*a2*a3*a4*a5*x1^2 - 8/625*a1^4*a5^2*x1^2 + 8/125*a0*a1^2*a2*a5^2*x1^2 - 2/25*a0^2*a2^2*a5^2*x1^2 sage: quintic.tau_covariant(as_form=True) Binary quadratic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.tau_covariant() 1/62500*a2^2*a3^4*x0^2 - 3/62500*a1*a3^5*x0^2 - 1/15625*a2^3*a3^2*a4*x0^2 + 1/6250*a1*a2*a3^3*a4*x0^2 + 3/6250*a0*a3^4*a4*x0^2 - 1/31250*a2^4*a4^2*x0^2 ... - 2/125*a0*a1*a2^2*a4*a5*x1^2 - 4/125*a0*a1^2*a3*a4*a5*x1^2 + 2/25*a0^2*a2*a3*a4*a5*x1^2 - 8/625*a1^4*a5^2*x1^2 + 8/125*a0*a1^2*a2*a5^2*x1^2 - 2/25*a0^2*a2^2*a5^2*x1^2 >>> quintic.tau_covariant(as_form=True) Binary quadratic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.tau_covariant() quintic.tau_covariant(as_form=True)
- theta_covariant(as_form=False)[source]¶
Return the covariant \(\theta\) of a binary quintic.
INPUT:
as_form– ifas_formisFalse, the result will be returned as polynomial (default). If it isTruethe result is returned as an object of the classAlgebraicForm.
OUTPUT:
The \(\theta\)-covariant of the binary quintic as polynomial or as binary form.
EXAMPLES:
sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: quintic.theta_covariant() -1/625000*a2^3*a3^5*x0^2 + 9/1250000*a1*a2*a3^6*x0^2 - 27/1250000*a0*a3^7*x0^2 + 3/250000*a2^4*a3^3*a4*x0^2 - 7/125000*a1*a2^2*a3^4*a4*x0^2 - 3/312500*a1^2*a3^5*a4*x0^2 ... + 6/625*a0^2*a1*a2^2*a4*a5^2*x1^2 + 24/625*a0^2*a1^2*a3*a4*a5^2*x1^2 - 12/125*a0^3*a2*a3*a4*a5^2*x1^2 + 8/625*a0*a1^4*a5^3*x1^2 - 8/125*a0^2*a1^2*a2*a5^3*x1^2 + 2/25*a0^3*a2^2*a5^3*x1^2 sage: quintic.theta_covariant(as_form=True) Binary quadratic given by ...
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> p = a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> quintic.theta_covariant() -1/625000*a2^3*a3^5*x0^2 + 9/1250000*a1*a2*a3^6*x0^2 - 27/1250000*a0*a3^7*x0^2 + 3/250000*a2^4*a3^3*a4*x0^2 - 7/125000*a1*a2^2*a3^4*a4*x0^2 - 3/312500*a1^2*a3^5*a4*x0^2 ... + 6/625*a0^2*a1*a2^2*a4*a5^2*x1^2 + 24/625*a0^2*a1^2*a3*a4*a5^2*x1^2 - 12/125*a0^3*a2*a3*a4*a5^2*x1^2 + 8/625*a0*a1^4*a5^3*x1^2 - 8/125*a0^2*a1^2*a2*a5^3*x1^2 + 2/25*a0^3*a2^2*a5^3*x1^2 >>> quintic.theta_covariant(as_form=True) Binary quadratic given by ...
R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[] p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) quintic.theta_covariant() quintic.theta_covariant(as_form=True)
- class sage.rings.invariants.invariant_theory.FormsBase(n, homogeneous, ring, variables)[source]¶
Bases:
SageObjectThe common base class of
AlgebraicFormandSeveralAlgebraicForms.This is an abstract base class to provide common methods. It does not make much sense to instantiate it.
- is_homogeneous()[source]¶
Return whether the forms were defined by homogeneous polynomials.
OUTPUT: boolean; whether the user originally defined the form via homogeneous variables
EXAMPLES:
sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) sage: quartic.is_homogeneous() True sage: quartic.form() x^2*y^2*t + x^4 + y^4 sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) sage: quartic.is_homogeneous() False sage: quartic.form() x^4 + x^2*t + 1
>>> from sage.all import * >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4) + t*x**Integer(2)*y**Integer(2), [x,y]) >>> quartic.is_homogeneous() True >>> quartic.form() x^2*y^2*t + x^4 + y^4 >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + Integer(1) + t*x**Integer(2), [x]) >>> quartic.is_homogeneous() False >>> quartic.form() x^4 + x^2*t + 1
R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) quartic.is_homogeneous() quartic.form() R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) quartic.is_homogeneous() quartic.form()
- ring()[source]¶
Return the polynomial ring.
OUTPUT:
A polynomial ring. This is where the defining polynomial(s) live. Note that the polynomials may be homogeneous or inhomogeneous, depending on how the user constructed the object.
EXAMPLES:
sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) sage: quartic.ring() Multivariate Polynomial Ring in x, y, t over Rational Field sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) sage: quartic.ring() Multivariate Polynomial Ring in x, y, t over Rational Field
>>> from sage.all import * >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4) + t*x**Integer(2)*y**Integer(2), [x,y]) >>> quartic.ring() Multivariate Polynomial Ring in x, y, t over Rational Field >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + Integer(1) + t*x**Integer(2), [x]) >>> quartic.ring() Multivariate Polynomial Ring in x, y, t over Rational Field
R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) quartic.ring() R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) quartic.ring()
- variables()[source]¶
Return the variables of the form.
OUTPUT:
A tuple of variables. If inhomogeneous notation is used for the defining polynomial then the last entry will be
None.EXAMPLES:
sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) sage: quartic.variables() (x, y) sage: R.<x,y,t> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) sage: quartic.variables() (x, None)
>>> from sage.all import * >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4) + t*x**Integer(2)*y**Integer(2), [x,y]) >>> quartic.variables() (x, y) >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + Integer(1) + t*x**Integer(2), [x]) >>> quartic.variables() (x, None)
R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y]) quartic.variables() R.<x,y,t> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x]) quartic.variables()
- class sage.rings.invariants.invariant_theory.InvariantTheoryFactory[source]¶
Bases:
objectFactory object for invariants of multilinear forms.
Use the invariant_theory object to construct algebraic forms. These can then be queried for invariant and covariants.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: invariant_theory.ternary_cubic(x^3 + y^3 + z^3) Ternary cubic with coefficients (1, 1, 1, 0, 0, 0, 0, 0, 0, 0) sage: invariant_theory.ternary_cubic(x^3 + y^3 + z^3).J_covariant() x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) Ternary cubic with coefficients (1, 1, 1, 0, 0, 0, 0, 0, 0, 0) >>> invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)).J_covariant() x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6
R.<x,y,z> = QQ[] invariant_theory.ternary_cubic(x^3 + y^3 + z^3) invariant_theory.ternary_cubic(x^3 + y^3 + z^3).J_covariant()
- binary_form_from_invariants(degree, invariants, variables=None, as_form=True, *args, **kwargs)[source]¶
Reconstruct a binary form from the values of its invariants.
INPUT:
degree– the degree of the binary forminvariants– list or tuple of values of the invariants of the binary formvariables– list or tuple of two variables that are used for the resulting form (only ifas_formisTrue). If no variables are provided, two abstract variablesxandzwill be used.as_form– boolean; ifFalse, the function will return a tuple of coefficients of a binary form
OUTPUT:
A binary form or a tuple of its coefficients, whose invariants are equal to the given
invariantsup to a scaling.EXAMPLES:
In the case of binary quadratics and cubics, the form is reconstructed based on the value of the discriminant. See also
binary_quadratic_coefficients_from_invariants()andbinary_cubic_coefficients_from_invariants(). These methods will always return the same result if the discriminant is nonzero:sage: discriminant = 1 sage: invariant_theory.binary_form_from_invariants(2, [discriminant]) Binary quadratic with coefficients (1, -1/4, 0) sage: invariant_theory.binary_form_from_invariants(3, [discriminant], as_form=false) (0, 1, -1, 0)
>>> from sage.all import * >>> discriminant = Integer(1) >>> invariant_theory.binary_form_from_invariants(Integer(2), [discriminant]) Binary quadratic with coefficients (1, -1/4, 0) >>> invariant_theory.binary_form_from_invariants(Integer(3), [discriminant], as_form=false) (0, 1, -1, 0)
discriminant = 1 invariant_theory.binary_form_from_invariants(2, [discriminant]) invariant_theory.binary_form_from_invariants(3, [discriminant], as_form=false)
For binary cubics, there is no class implemented yet, so
as_form=Truewill yield aNotImplementedError:sage: invariant_theory.binary_form_from_invariants(3, [discriminant]) Traceback (most recent call last): ... NotImplementedError: no class for binary cubics implemented
>>> from sage.all import * >>> invariant_theory.binary_form_from_invariants(Integer(3), [discriminant]) Traceback (most recent call last): ... NotImplementedError: no class for binary cubics implemented
invariant_theory.binary_form_from_invariants(3, [discriminant])
For binary quintics, the three Clebsch invariants of the form should be provided to reconstruct the form. For more details about these invariants, see
clebsch_invariants():sage: invariants = [1, 0, 0] sage: invariant_theory.binary_form_from_invariants(5, invariants) Binary quintic with coefficients (1, 0, 0, 0, 0, 1)
>>> from sage.all import * >>> invariants = [Integer(1), Integer(0), Integer(0)] >>> invariant_theory.binary_form_from_invariants(Integer(5), invariants) Binary quintic with coefficients (1, 0, 0, 0, 0, 1)
invariants = [1, 0, 0] invariant_theory.binary_form_from_invariants(5, invariants)
An optional
scalingargument may be provided in order to scale the resulting quintic. For more details, seebinary_quintic_coefficients_from_invariants():sage: invariants = [3, 4, 7] sage: invariant_theory.binary_form_from_invariants(5, invariants) Binary quintic with coefficients (-37725479487783/1048576, 565882192316745/8388608, 0, 1033866765362693115/67108864, 12849486940936328715/268435456, -23129076493685391687/2147483648) sage: invariant_theory.binary_form_from_invariants(5, invariants, ....: scaling='normalized') Binary quintic with coefficients (24389/892616806656, 4205/11019960576, 0, 1015/209952, -145/1296, -3/16) sage: invariant_theory.binary_form_from_invariants(5, invariants, ....: scaling='coprime') Binary quintic with coefficients (-2048, 3840, 0, 876960, 2724840, -613089)
>>> from sage.all import * >>> invariants = [Integer(3), Integer(4), Integer(7)] >>> invariant_theory.binary_form_from_invariants(Integer(5), invariants) Binary quintic with coefficients (-37725479487783/1048576, 565882192316745/8388608, 0, 1033866765362693115/67108864, 12849486940936328715/268435456, -23129076493685391687/2147483648) >>> invariant_theory.binary_form_from_invariants(Integer(5), invariants, ... scaling='normalized') Binary quintic with coefficients (24389/892616806656, 4205/11019960576, 0, 1015/209952, -145/1296, -3/16) >>> invariant_theory.binary_form_from_invariants(Integer(5), invariants, ... scaling='coprime') Binary quintic with coefficients (-2048, 3840, 0, 876960, 2724840, -613089)
invariants = [3, 4, 7] invariant_theory.binary_form_from_invariants(5, invariants) invariant_theory.binary_form_from_invariants(5, invariants, scaling='normalized') invariant_theory.binary_form_from_invariants(5, invariants, scaling='coprime')The invariants can also be computed using the invariants of a given binary quintic. The resulting form has the same invariants up to scaling, is \(GL(2,\QQ)\)-equivalent to the provided form and hence has the same canonical form (see
canonical_form()):sage: R.<x0, x1> = QQ[] sage: p = 3*x1^5 + 6*x1^4*x0 + 3*x1^3*x0^2 + 4*x1^2*x0^3 - 5*x1*x0^4 + 4*x0^5 sage: quintic = invariant_theory.binary_quintic(p, x0, x1) sage: invariants = quintic.clebsch_invariants(as_tuple=True) sage: newquintic = invariant_theory.binary_form_from_invariants( ....: 5, invariants, variables=quintic.variables()) sage: newquintic Binary quintic with coefficients (9592267437341790539005557/244140625000000, 2149296928207625556323004064707/610351562500000000, 11149651890347700974453304786783/76293945312500000, 122650775751894638395648891202734239/47683715820312500000, 323996630945706528474286334593218447/11920928955078125000, 1504506503644608395841632538558481466127/14901161193847656250000) sage: quintic.canonical_form() == newquintic.canonical_form() True
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(3)*x1**Integer(5) + Integer(6)*x1**Integer(4)*x0 + Integer(3)*x1**Integer(3)*x0**Integer(2) + Integer(4)*x1**Integer(2)*x0**Integer(3) - Integer(5)*x1*x0**Integer(4) + Integer(4)*x0**Integer(5) >>> quintic = invariant_theory.binary_quintic(p, x0, x1) >>> invariants = quintic.clebsch_invariants(as_tuple=True) >>> newquintic = invariant_theory.binary_form_from_invariants( ... Integer(5), invariants, variables=quintic.variables()) >>> newquintic Binary quintic with coefficients (9592267437341790539005557/244140625000000, 2149296928207625556323004064707/610351562500000000, 11149651890347700974453304786783/76293945312500000, 122650775751894638395648891202734239/47683715820312500000, 323996630945706528474286334593218447/11920928955078125000, 1504506503644608395841632538558481466127/14901161193847656250000) >>> quintic.canonical_form() == newquintic.canonical_form() True
R.<x0, x1> = QQ[] p = 3*x1^5 + 6*x1^4*x0 + 3*x1^3*x0^2 + 4*x1^2*x0^3 - 5*x1*x0^4 + 4*x0^5 quintic = invariant_theory.binary_quintic(p, x0, x1) invariants = quintic.clebsch_invariants(as_tuple=True) newquintic = invariant_theory.binary_form_from_invariants( 5, invariants, variables=quintic.variables()) newquintic quintic.canonical_form() == newquintic.canonical_form()For binary forms of other degrees, no reconstruction has been implemented yet. For forms of degree 6, see Issue #26462:
sage: invariant_theory.binary_form_from_invariants(6, invariants) Traceback (most recent call last): ... NotImplementedError: no reconstruction for binary forms of degree 6 implemented
>>> from sage.all import * >>> invariant_theory.binary_form_from_invariants(Integer(6), invariants) Traceback (most recent call last): ... NotImplementedError: no reconstruction for binary forms of degree 6 implemented
invariant_theory.binary_form_from_invariants(6, invariants)
- binary_quadratic(quadratic, *args)[source]¶
Invariant theory of a quadratic in two variables.
INPUT:
quadratic– a quadratic formx,y– the homogeneous variables. IfyisNone, the quadratic is assumed to be inhomogeneous.
REFERENCES:
EXAMPLES:
sage: R.<x,y> = QQ[] sage: invariant_theory.binary_quadratic(x^2 + y^2) Binary quadratic with coefficients (1, 1, 0) sage: T.<t> = QQ[] sage: invariant_theory.binary_quadratic(t^2 + 2*t + 1, [t]) Binary quadratic with coefficients (1, 1, 2)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> invariant_theory.binary_quadratic(x**Integer(2) + y**Integer(2)) Binary quadratic with coefficients (1, 1, 0) >>> T = QQ['t']; (t,) = T._first_ngens(1) >>> invariant_theory.binary_quadratic(t**Integer(2) + Integer(2)*t + Integer(1), [t]) Binary quadratic with coefficients (1, 1, 2)
R.<x,y> = QQ[] invariant_theory.binary_quadratic(x^2 + y^2) T.<t> = QQ[] invariant_theory.binary_quadratic(t^2 + 2*t + 1, [t])
- binary_quartic(quartic, *args, **kwds)[source]¶
Invariant theory of a quartic in two variables.
The algebra of invariants of a quartic form is generated by invariants \(i\), \(j\) of degrees 2, 3. This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series \(E_4\) (see
EisensteinD()) and \(E_6\) (seeEisensteinE()). The algebra of covariants is generated by these two invariants together with the form \(f\) of degree 1 and order 4, the Hessian \(g\) (seeg_covariant()) of degree 2 and order 4, and a covariant \(h\) (seeh_covariant()) of degree 3 and order 6. They are related by a syzygy\[j f^3 - g f^2 i + 4 g^3 + h^2 = 0\]of degree 6 and order 12.
INPUT:
quartic– a quarticx,y– the homogeneous variables. IfyisNone, the quartic is assumed to be inhomogeneous.
REFERENCES:
EXAMPLES:
sage: R.<x,y> = QQ[] sage: quartic = invariant_theory.binary_quartic(x^4 + y^4) sage: quartic Binary quartic with coefficients (1, 0, 0, 0, 1) sage: type(quartic) <class 'sage.rings.invariants.invariant_theory.BinaryQuartic'>
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quartic = invariant_theory.binary_quartic(x**Integer(4) + y**Integer(4)) >>> quartic Binary quartic with coefficients (1, 0, 0, 0, 1) >>> type(quartic) <class 'sage.rings.invariants.invariant_theory.BinaryQuartic'>
R.<x,y> = QQ[] quartic = invariant_theory.binary_quartic(x^4 + y^4) quartic type(quartic)
- binary_quintic(quintic, *args, **kwds)[source]¶
Create a binary quintic for computing invariants.
A binary quintic is a homogeneous polynomial of degree 5 in two variables. The algebra of invariants of a binary quintic is generated by the invariants \(A\), \(B\) and \(C\) of respective degrees 4, 8 and 12 (see
A_invariant(),B_invariant()andC_invariant()).INPUT:
quintic– a homogeneous polynomial of degree five in two variables or a (possibly inhomogeneous) polynomial of degree at most five in one variable.*args– the two homogeneous variables. If only one variable is given, the polynomialquinticis assumed to be univariate. If no variables are given, they are guessed.
REFERENCES:
EXAMPLES:
If no variables are provided, they will be guessed:
sage: R.<x,y> = QQ[] sage: quintic = invariant_theory.binary_quintic(x^5 + y^5) sage: quintic Binary quintic with coefficients (1, 0, 0, 0, 0, 1)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + y**Integer(5)) >>> quintic Binary quintic with coefficients (1, 0, 0, 0, 0, 1)
R.<x,y> = QQ[] quintic = invariant_theory.binary_quintic(x^5 + y^5) quintic
If only one variable is given, the quintic is the homogenisation of the provided polynomial:
sage: quintic = invariant_theory.binary_quintic(x^5 + y^5, x) sage: quintic Binary quintic with coefficients (y^5, 0, 0, 0, 0, 1) sage: quintic.is_homogeneous() False
>>> from sage.all import * >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + y**Integer(5), x) >>> quintic Binary quintic with coefficients (y^5, 0, 0, 0, 0, 1) >>> quintic.is_homogeneous() False
quintic = invariant_theory.binary_quintic(x^5 + y^5, x) quintic quintic.is_homogeneous()
If the polynomial has three or more variables, the variables should be specified:
sage: R.<x,y,z> = QQ[] sage: quintic = invariant_theory.binary_quintic(x^5 + z*y^5) Traceback (most recent call last): ... ValueError: need 2 or 1 variables, got (x, y, z) sage: quintic = invariant_theory.binary_quintic(x^5 + z*y^5, x, y) sage: quintic Binary quintic with coefficients (z, 0, 0, 0, 0, 1) sage: type(quintic) <class 'sage.rings.invariants.invariant_theory.BinaryQuintic'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + z*y**Integer(5)) Traceback (most recent call last): ... ValueError: need 2 or 1 variables, got (x, y, z) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + z*y**Integer(5), x, y) >>> quintic Binary quintic with coefficients (z, 0, 0, 0, 0, 1) >>> type(quintic) <class 'sage.rings.invariants.invariant_theory.BinaryQuintic'>
R.<x,y,z> = QQ[] quintic = invariant_theory.binary_quintic(x^5 + z*y^5) quintic = invariant_theory.binary_quintic(x^5 + z*y^5, x, y) quintic type(quintic)
- inhomogeneous_quadratic_form(polynomial, *args)[source]¶
Invariants of an inhomogeneous quadratic form.
INPUT:
polynomial– an inhomogeneous quadratic form*args– the variables as multiple arguments, or as a single list/tuple
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: quadratic = x^2 + 2*y^2 + 3*x*y + 4*x + 5*y + 6 sage: inv3 = invariant_theory.inhomogeneous_quadratic_form(quadratic) sage: type(inv3) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'> sage: inv4 = invariant_theory.inhomogeneous_quadratic_form(x^2 + y^2 + z^2) sage: type(inv4) <class 'sage.rings.invariants.invariant_theory.QuadraticForm'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quadratic = x**Integer(2) + Integer(2)*y**Integer(2) + Integer(3)*x*y + Integer(4)*x + Integer(5)*y + Integer(6) >>> inv3 = invariant_theory.inhomogeneous_quadratic_form(quadratic) >>> type(inv3) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'> >>> inv4 = invariant_theory.inhomogeneous_quadratic_form(x**Integer(2) + y**Integer(2) + z**Integer(2)) >>> type(inv4) <class 'sage.rings.invariants.invariant_theory.QuadraticForm'>
R.<x,y,z> = QQ[] quadratic = x^2 + 2*y^2 + 3*x*y + 4*x + 5*y + 6 inv3 = invariant_theory.inhomogeneous_quadratic_form(quadratic) type(inv3) inv4 = invariant_theory.inhomogeneous_quadratic_form(x^2 + y^2 + z^2) type(inv4)
- quadratic_form(polynomial, *args)[source]¶
Invariants of a homogeneous quadratic form.
INPUT:
polynomial– a homogeneous or inhomogeneous quadratic form*args– the variables as multiple arguments, or as a single list/tuple. If the last argument isNone, the cubic is assumed to be inhomogeneous.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: quadratic = x^2 + y^2 + z^2 sage: inv = invariant_theory.quadratic_form(quadratic) sage: type(inv) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quadratic = x**Integer(2) + y**Integer(2) + z**Integer(2) >>> inv = invariant_theory.quadratic_form(quadratic) >>> type(inv) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>
R.<x,y,z> = QQ[] quadratic = x^2 + y^2 + z^2 inv = invariant_theory.quadratic_form(quadratic) type(inv)
If some of the ring variables are to be treated as coefficients you need to specify the polynomial variables:
sage: R.<x,y,z, a,b> = QQ[] sage: quadratic = a*x^2 + b*y^2 + z^2 + 2*y*z sage: invariant_theory.quadratic_form(quadratic, x,y,z) Ternary quadratic with coefficients (a, b, 1, 0, 0, 2) sage: invariant_theory.quadratic_form(quadratic, [x,y,z]) # alternate syntax Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
>>> from sage.all import * >>> R = QQ['x, y, z, a, b']; (x, y, z, a, b,) = R._first_ngens(5) >>> quadratic = a*x**Integer(2) + b*y**Integer(2) + z**Integer(2) + Integer(2)*y*z >>> invariant_theory.quadratic_form(quadratic, x,y,z) Ternary quadratic with coefficients (a, b, 1, 0, 0, 2) >>> invariant_theory.quadratic_form(quadratic, [x,y,z]) # alternate syntax Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
R.<x,y,z, a,b> = QQ[] quadratic = a*x^2 + b*y^2 + z^2 + 2*y*z invariant_theory.quadratic_form(quadratic, x,y,z) invariant_theory.quadratic_form(quadratic, [x,y,z]) # alternate syntax
Inhomogeneous quadratic forms (see also
inhomogeneous_quadratic_form()) can be specified by passingNoneas the last variable:sage: inhom = quadratic.subs(z=1) sage: invariant_theory.quadratic_form(inhom, x,y,None) Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
>>> from sage.all import * >>> inhom = quadratic.subs(z=Integer(1)) >>> invariant_theory.quadratic_form(inhom, x,y,None) Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
inhom = quadratic.subs(z=1) invariant_theory.quadratic_form(inhom, x,y,None)
- quaternary_biquadratic(quadratic1, quadratic2, *args, **kwds)[source]¶
Invariants of two quadratics in four variables.
INPUT:
quadratic1,quadratic2– two polynomials. Either homogeneous quadratic in 4 homogeneous variables, or inhomogeneous quadratic in 3 variables.w,x,y,z– the variables. IfzisNone, the quadratics are assumed to be inhomogeneous.
EXAMPLES:
sage: R.<w,x,y,z> = QQ[] sage: q1 = w^2 + x^2 + y^2 + z^2 sage: q2 = w*x + y*z sage: inv = invariant_theory.quaternary_biquadratic(q1, q2) sage: type(inv) <class 'sage.rings.invariants.invariant_theory.TwoQuaternaryQuadratics'>
>>> from sage.all import * >>> R = QQ['w, x, y, z']; (w, x, y, z,) = R._first_ngens(4) >>> q1 = w**Integer(2) + x**Integer(2) + y**Integer(2) + z**Integer(2) >>> q2 = w*x + y*z >>> inv = invariant_theory.quaternary_biquadratic(q1, q2) >>> type(inv) <class 'sage.rings.invariants.invariant_theory.TwoQuaternaryQuadratics'>
R.<w,x,y,z> = QQ[] q1 = w^2 + x^2 + y^2 + z^2 q2 = w*x + y*z inv = invariant_theory.quaternary_biquadratic(q1, q2) type(inv)
Distance between two spheres [Sal1958], [Sal1965]
sage: R.<x,y,z, a,b,c, r1,r2> = QQ[] sage: S1 = -r1^2 + x^2 + y^2 + z^2 sage: S2 = -r2^2 + (x-a)^2 + (y-b)^2 + (z-c)^2 sage: inv = invariant_theory.quaternary_biquadratic(S1, S2, [x, y, z]) sage: inv.Delta_invariant() -r1^2 sage: inv.Delta_prime_invariant() -r2^2 sage: inv.Theta_invariant() a^2 + b^2 + c^2 - 3*r1^2 - r2^2 sage: inv.Theta_prime_invariant() a^2 + b^2 + c^2 - r1^2 - 3*r2^2 sage: inv.Phi_invariant() 2*a^2 + 2*b^2 + 2*c^2 - 3*r1^2 - 3*r2^2 sage: inv.J_covariant() 0
>>> from sage.all import * >>> R = QQ['x, y, z, a, b, c, r1, r2']; (x, y, z, a, b, c, r1, r2,) = R._first_ngens(8) >>> S1 = -r1**Integer(2) + x**Integer(2) + y**Integer(2) + z**Integer(2) >>> S2 = -r2**Integer(2) + (x-a)**Integer(2) + (y-b)**Integer(2) + (z-c)**Integer(2) >>> inv = invariant_theory.quaternary_biquadratic(S1, S2, [x, y, z]) >>> inv.Delta_invariant() -r1^2 >>> inv.Delta_prime_invariant() -r2^2 >>> inv.Theta_invariant() a^2 + b^2 + c^2 - 3*r1^2 - r2^2 >>> inv.Theta_prime_invariant() a^2 + b^2 + c^2 - r1^2 - 3*r2^2 >>> inv.Phi_invariant() 2*a^2 + 2*b^2 + 2*c^2 - 3*r1^2 - 3*r2^2 >>> inv.J_covariant() 0
R.<x,y,z, a,b,c, r1,r2> = QQ[] S1 = -r1^2 + x^2 + y^2 + z^2 S2 = -r2^2 + (x-a)^2 + (y-b)^2 + (z-c)^2 inv = invariant_theory.quaternary_biquadratic(S1, S2, [x, y, z]) inv.Delta_invariant() inv.Delta_prime_invariant() inv.Theta_invariant() inv.Theta_prime_invariant() inv.Phi_invariant() inv.J_covariant()
- quaternary_quadratic(quadratic, *args)[source]¶
Invariant theory of a quadratic in four variables.
INPUT:
quadratic– a quadratic formw,x,y,z– the homogeneous variables. IfzisNone, the quadratic is assumed to be inhomogeneous.
REFERENCES:
EXAMPLES:
sage: R.<w,x,y,z> = QQ[] sage: invariant_theory.quaternary_quadratic(w^2 + x^2 + y^2 + z^2) Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0) sage: R.<x,y,z> = QQ[] sage: invariant_theory.quaternary_quadratic(1 + x^2 + y^2 + z^2) Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)
>>> from sage.all import * >>> R = QQ['w, x, y, z']; (w, x, y, z,) = R._first_ngens(4) >>> invariant_theory.quaternary_quadratic(w**Integer(2) + x**Integer(2) + y**Integer(2) + z**Integer(2)) Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0) >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> invariant_theory.quaternary_quadratic(Integer(1) + x**Integer(2) + y**Integer(2) + z**Integer(2)) Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)
R.<w,x,y,z> = QQ[] invariant_theory.quaternary_quadratic(w^2 + x^2 + y^2 + z^2) R.<x,y,z> = QQ[] invariant_theory.quaternary_quadratic(1 + x^2 + y^2 + z^2)
- ternary_biquadratic(quadratic1, quadratic2, *args, **kwds)[source]¶
Invariants of two quadratics in three variables.
INPUT:
quadratic1,quadratic2– two polynomials. Either homogeneous quadratic in 3 homogeneous variables, or inhomogeneous quadratic in 2 variables.x,y,z– the variables. IfzisNone, the quadratics are assumed to be inhomogeneous.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: q1 = x^2 + y^2 + z^2 sage: q2 = x*y + y*z + x*z sage: inv = invariant_theory.ternary_biquadratic(q1, q2) sage: type(inv) <class 'sage.rings.invariants.invariant_theory.TwoTernaryQuadratics'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> q1 = x**Integer(2) + y**Integer(2) + z**Integer(2) >>> q2 = x*y + y*z + x*z >>> inv = invariant_theory.ternary_biquadratic(q1, q2) >>> type(inv) <class 'sage.rings.invariants.invariant_theory.TwoTernaryQuadratics'>
R.<x,y,z> = QQ[] q1 = x^2 + y^2 + z^2 q2 = x*y + y*z + x*z inv = invariant_theory.ternary_biquadratic(q1, q2) type(inv)
Distance between two circles:
sage: R.<x,y, a,b, r1,r2> = QQ[] sage: S1 = -r1^2 + x^2 + y^2 sage: S2 = -r2^2 + (x-a)^2 + (y-b)^2 sage: inv = invariant_theory.ternary_biquadratic(S1, S2, [x, y]) sage: inv.Delta_invariant() -r1^2 sage: inv.Delta_prime_invariant() -r2^2 sage: inv.Theta_invariant() a^2 + b^2 - 2*r1^2 - r2^2 sage: inv.Theta_prime_invariant() a^2 + b^2 - r1^2 - 2*r2^2 sage: inv.F_covariant() 2*x^2*a^2 + y^2*a^2 - 2*x*a^3 + a^4 + 2*x*y*a*b - 2*y*a^2*b + x^2*b^2 + 2*y^2*b^2 - 2*x*a*b^2 + 2*a^2*b^2 - 2*y*b^3 + b^4 - 2*x^2*r1^2 - 2*y^2*r1^2 + 2*x*a*r1^2 - 2*a^2*r1^2 + 2*y*b*r1^2 - 2*b^2*r1^2 + r1^4 - 2*x^2*r2^2 - 2*y^2*r2^2 + 2*x*a*r2^2 - 2*a^2*r2^2 + 2*y*b*r2^2 - 2*b^2*r2^2 + 2*r1^2*r2^2 + r2^4 sage: inv.J_covariant() -8*x^2*y*a^3 + 8*x*y*a^4 + 8*x^3*a^2*b - 16*x*y^2*a^2*b - 8*x^2*a^3*b + 8*y^2*a^3*b + 16*x^2*y*a*b^2 - 8*y^3*a*b^2 + 8*x*y^2*b^3 - 8*x^2*a*b^3 + 8*y^2*a*b^3 - 8*x*y*b^4 + 8*x*y*a^2*r1^2 - 8*y*a^3*r1^2 - 8*x^2*a*b*r1^2 + 8*y^2*a*b*r1^2 + 8*x*a^2*b*r1^2 - 8*x*y*b^2*r1^2 - 8*y*a*b^2*r1^2 + 8*x*b^3*r1^2 - 8*x*y*a^2*r2^2 + 8*x^2*a*b*r2^2 - 8*y^2*a*b*r2^2 + 8*x*y*b^2*r2^2
>>> from sage.all import * >>> R = QQ['x, y, a, b, r1, r2']; (x, y, a, b, r1, r2,) = R._first_ngens(6) >>> S1 = -r1**Integer(2) + x**Integer(2) + y**Integer(2) >>> S2 = -r2**Integer(2) + (x-a)**Integer(2) + (y-b)**Integer(2) >>> inv = invariant_theory.ternary_biquadratic(S1, S2, [x, y]) >>> inv.Delta_invariant() -r1^2 >>> inv.Delta_prime_invariant() -r2^2 >>> inv.Theta_invariant() a^2 + b^2 - 2*r1^2 - r2^2 >>> inv.Theta_prime_invariant() a^2 + b^2 - r1^2 - 2*r2^2 >>> inv.F_covariant() 2*x^2*a^2 + y^2*a^2 - 2*x*a^3 + a^4 + 2*x*y*a*b - 2*y*a^2*b + x^2*b^2 + 2*y^2*b^2 - 2*x*a*b^2 + 2*a^2*b^2 - 2*y*b^3 + b^4 - 2*x^2*r1^2 - 2*y^2*r1^2 + 2*x*a*r1^2 - 2*a^2*r1^2 + 2*y*b*r1^2 - 2*b^2*r1^2 + r1^4 - 2*x^2*r2^2 - 2*y^2*r2^2 + 2*x*a*r2^2 - 2*a^2*r2^2 + 2*y*b*r2^2 - 2*b^2*r2^2 + 2*r1^2*r2^2 + r2^4 >>> inv.J_covariant() -8*x^2*y*a^3 + 8*x*y*a^4 + 8*x^3*a^2*b - 16*x*y^2*a^2*b - 8*x^2*a^3*b + 8*y^2*a^3*b + 16*x^2*y*a*b^2 - 8*y^3*a*b^2 + 8*x*y^2*b^3 - 8*x^2*a*b^3 + 8*y^2*a*b^3 - 8*x*y*b^4 + 8*x*y*a^2*r1^2 - 8*y*a^3*r1^2 - 8*x^2*a*b*r1^2 + 8*y^2*a*b*r1^2 + 8*x*a^2*b*r1^2 - 8*x*y*b^2*r1^2 - 8*y*a*b^2*r1^2 + 8*x*b^3*r1^2 - 8*x*y*a^2*r2^2 + 8*x^2*a*b*r2^2 - 8*y^2*a*b*r2^2 + 8*x*y*b^2*r2^2
R.<x,y, a,b, r1,r2> = QQ[] S1 = -r1^2 + x^2 + y^2 S2 = -r2^2 + (x-a)^2 + (y-b)^2 inv = invariant_theory.ternary_biquadratic(S1, S2, [x, y]) inv.Delta_invariant() inv.Delta_prime_invariant() inv.Theta_invariant() inv.Theta_prime_invariant() inv.F_covariant() inv.J_covariant()
- ternary_cubic(cubic, *args, **kwds)[source]¶
Invariants of a cubic in three variables.
The algebra of invariants of a ternary cubic under \(SL_3(\CC)\) is a polynomial algebra generated by two invariants \(S\) (see
S_invariant()) and T (seeT_invariant()) of degrees 4 and 6, called Aronhold invariants.The ring of covariants is given as follows. The identity covariant U of a ternary cubic has degree 1 and order 3. The Hessian \(H\) (see
Hessian()) is a covariant of ternary cubics of degree 3 and order 3. There is a covariant \(\Theta\) (seeTheta_covariant()) of ternary cubics of degree 8 and order 6 that vanishes on points \(x\) lying on the Salmon conic of the polar of \(x\) with respect to the curve and its Hessian curve. The Brioschi covariant \(J\) (seeJ_covariant()) is the Jacobian of \(U\), \(\Theta\), and \(H\) of degree 12, order 9. The algebra of covariants of a ternary cubic is generated over the ring of invariants by \(U\), \(\Theta\), \(H\), and \(J\), with a relation\[\begin{split}\begin{split} J^2 =& 4 \Theta^3 + T U^2 \Theta^2 + \Theta (-4 S^3 U^4 + 2 S T U^3 H - 72 S^2 U^2 H^2 \\ & - 18 T U H^3 + 108 S H^4) -16 S^4 U^5 H - 11 S^2 T U^4 H^2 \\ & -4 T^2 U^3 H^3 +54 S T U^2 H^4 -432 S^2 U H^5 -27 T H^6 \end{split}\end{split}\]REFERENCES:
INPUT:
cubic– a homogeneous cubic in 3 homogeneous variables, or an inhomogeneous cubic in 2 variablesx,y,z– the variables. IfzisNone, the cubic is assumed to be inhomogeneous.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) sage: type(cubic) <class 'sage.rings.invariants.invariant_theory.TernaryCubic'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) >>> type(cubic) <class 'sage.rings.invariants.invariant_theory.TernaryCubic'>
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) type(cubic)
- ternary_quadratic(quadratic, *args, **kwds)[source]¶
Invariants of a quadratic in three variables.
INPUT:
quadratic– a homogeneous quadratic in 3 homogeneous variables, or an inhomogeneous quadratic in 2 variablesx,y,z– the variables. IfzisNone, the quadratic is assumed to be inhomogeneous.
REFERENCES:
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: invariant_theory.ternary_quadratic(x^2 + y^2 + z^2) Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0) sage: T.<u, v> = QQ[] sage: invariant_theory.ternary_quadratic(1 + u^2 + v^2) Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0) sage: quadratic = x^2 + y^2 + z^2 sage: inv = invariant_theory.ternary_quadratic(quadratic) sage: type(inv) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> invariant_theory.ternary_quadratic(x**Integer(2) + y**Integer(2) + z**Integer(2)) Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0) >>> T = QQ['u, v']; (u, v,) = T._first_ngens(2) >>> invariant_theory.ternary_quadratic(Integer(1) + u**Integer(2) + v**Integer(2)) Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0) >>> quadratic = x**Integer(2) + y**Integer(2) + z**Integer(2) >>> inv = invariant_theory.ternary_quadratic(quadratic) >>> type(inv) <class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>
R.<x,y,z> = QQ[] invariant_theory.ternary_quadratic(x^2 + y^2 + z^2) T.<u, v> = QQ[] invariant_theory.ternary_quadratic(1 + u^2 + v^2) quadratic = x^2 + y^2 + z^2 inv = invariant_theory.ternary_quadratic(quadratic) type(inv)
- class sage.rings.invariants.invariant_theory.QuadraticForm(n, d, polynomial, *args)[source]¶
Bases:
AlgebraicFormInvariant theory of a multivariate quadratic form.
You should use the
invariant_theoryfactory object to construct instances of this class. Seequadratic_form()for details.- as_QuadraticForm()[source]¶
Convert into a
QuadraticForm.OUTPUT:
Sage has a special quadratic forms subsystem. This method converts
selfinto thisQuadraticFormrepresentation.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: p = x^2 + y^2 + z^2 + 2*x*y + 3*x*z sage: quadratic = invariant_theory.ternary_quadratic(p) sage: matrix(quadratic) [ 1 1 3/2] [ 1 1 0] [3/2 0 1] sage: quadratic.as_QuadraticForm() Quadratic form in 3 variables over Multivariate Polynomial Ring in x, y, z over Rational Field with coefficients: [ 1 2 3 ] [ * 1 0 ] [ * * 1 ] sage: _.polynomial('X,Y,Z') X^2 + 2*X*Y + Y^2 + 3*X*Z + Z^2
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> p = x**Integer(2) + y**Integer(2) + z**Integer(2) + Integer(2)*x*y + Integer(3)*x*z >>> quadratic = invariant_theory.ternary_quadratic(p) >>> matrix(quadratic) [ 1 1 3/2] [ 1 1 0] [3/2 0 1] >>> quadratic.as_QuadraticForm() Quadratic form in 3 variables over Multivariate Polynomial Ring in x, y, z over Rational Field with coefficients: [ 1 2 3 ] [ * 1 0 ] [ * * 1 ] >>> _.polynomial('X,Y,Z') X^2 + 2*X*Y + Y^2 + 3*X*Z + Z^2
R.<x,y,z> = QQ[] p = x^2 + y^2 + z^2 + 2*x*y + 3*x*z quadratic = invariant_theory.ternary_quadratic(p) matrix(quadratic) quadratic.as_QuadraticForm() _.polynomial('X,Y,Z')
- coeffs()[source]¶
The coefficients of a quadratic form.
Given
\[f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} a_{jk} x_j x_k\]this function returns \(a = (a_0, \dots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)
EXAMPLES:
sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[] sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv Ternary quadratic with coefficients (a, b, c, d, e, f) sage: inv.coeffs() (a, b, c, d, e, f) sage: inv.scaled_coeffs() (a, b, c, 1/2*d, 1/2*e, 1/2*f)
>>> from sage.all import * >>> R = QQ['a, b, c, d, e, f, g, x, y, z']; (a, b, c, d, e, f, g, x, y, z,) = R._first_ngens(10) >>> p = a*x**Integer(2) + b*y**Integer(2) + c*z**Integer(2) + d*x*y + e*x*z + f*y*z >>> inv = invariant_theory.quadratic_form(p, x,y,z); inv Ternary quadratic with coefficients (a, b, c, d, e, f) >>> inv.coeffs() (a, b, c, d, e, f) >>> inv.scaled_coeffs() (a, b, c, 1/2*d, 1/2*e, 1/2*f)
R.<a,b,c,d,e,f,g, x,y,z> = QQ[] p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z inv = invariant_theory.quadratic_form(p, x,y,z); inv inv.coeffs() inv.scaled_coeffs()
- discriminant()[source]¶
Return the discriminant of the quadratic form.
Up to an overall constant factor, this is just the determinant of the defining matrix, see
matrix(). For a quadratic form in \(n\) variables, the overall constant is \(2^{n-1}\) if \(n\) is odd and \((-1)^{n/2} 2^n\) if \(n\) is even.EXAMPLES:
sage: R.<a,b,c, x,y> = QQ[] sage: p = a*x^2 + b*x*y + c*y^2 sage: quadratic = invariant_theory.quadratic_form(p, x,y) sage: quadratic.discriminant() b^2 - 4*a*c sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[] sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z sage: quadratic = invariant_theory.quadratic_form(p, x,y,z) sage: quadratic.discriminant() 4*a*b*c - c*d^2 - b*e^2 + d*e*f - a*f^2
>>> from sage.all import * >>> R = QQ['a, b, c, x, y']; (a, b, c, x, y,) = R._first_ngens(5) >>> p = a*x**Integer(2) + b*x*y + c*y**Integer(2) >>> quadratic = invariant_theory.quadratic_form(p, x,y) >>> quadratic.discriminant() b^2 - 4*a*c >>> R = QQ['a, b, c, d, e, f, g, x, y, z']; (a, b, c, d, e, f, g, x, y, z,) = R._first_ngens(10) >>> p = a*x**Integer(2) + b*y**Integer(2) + c*z**Integer(2) + d*x*y + e*x*z + f*y*z >>> quadratic = invariant_theory.quadratic_form(p, x,y,z) >>> quadratic.discriminant() 4*a*b*c - c*d^2 - b*e^2 + d*e*f - a*f^2
R.<a,b,c, x,y> = QQ[] p = a*x^2 + b*x*y + c*y^2 quadratic = invariant_theory.quadratic_form(p, x,y) quadratic.discriminant() R.<a,b,c,d,e,f,g, x,y,z> = QQ[] p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z quadratic = invariant_theory.quadratic_form(p, x,y,z) quadratic.discriminant()
- dual()[source]¶
Return the dual quadratic form.
OUTPUT:
A new quadratic form (with the same number of variables) defined by the adjoint matrix.
EXAMPLES:
sage: R.<a,b,c,x,y,z> = QQ[] sage: cubic = x^2+y^2+z^2 sage: quadratic = invariant_theory.ternary_quadratic(a*x^2+b*y^2+c*z^2, [x,y,z]) sage: quadratic.form() a*x^2 + b*y^2 + c*z^2 sage: quadratic.dual().form() b*c*x^2 + a*c*y^2 + a*b*z^2 sage: R.<x,y,z, t> = QQ[] sage: cubic = x^2+y^2+z^2 sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2 + t*x*y, [x,y,z]) sage: quadratic.dual() Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0) sage: R.<x,y, t> = QQ[] sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+1 + t*x*y, [x,y]) sage: quadratic.dual() Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)
>>> from sage.all import * >>> R = QQ['a, b, c, x, y, z']; (a, b, c, x, y, z,) = R._first_ngens(6) >>> cubic = x**Integer(2)+y**Integer(2)+z**Integer(2) >>> quadratic = invariant_theory.ternary_quadratic(a*x**Integer(2)+b*y**Integer(2)+c*z**Integer(2), [x,y,z]) >>> quadratic.form() a*x^2 + b*y^2 + c*z^2 >>> quadratic.dual().form() b*c*x^2 + a*c*y^2 + a*b*z^2 >>> R = QQ['x, y, z, t']; (x, y, z, t,) = R._first_ngens(4) >>> cubic = x**Integer(2)+y**Integer(2)+z**Integer(2) >>> quadratic = invariant_theory.ternary_quadratic(x**Integer(2)+y**Integer(2)+z**Integer(2) + t*x*y, [x,y,z]) >>> quadratic.dual() Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0) >>> R = QQ['x, y, t']; (x, y, t,) = R._first_ngens(3) >>> quadratic = invariant_theory.ternary_quadratic(x**Integer(2)+y**Integer(2)+Integer(1) + t*x*y, [x,y]) >>> quadratic.dual() Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)
R.<a,b,c,x,y,z> = QQ[] cubic = x^2+y^2+z^2 quadratic = invariant_theory.ternary_quadratic(a*x^2+b*y^2+c*z^2, [x,y,z]) quadratic.form() quadratic.dual().form() R.<x,y,z, t> = QQ[] cubic = x^2+y^2+z^2 quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2 + t*x*y, [x,y,z]) quadratic.dual() R.<x,y, t> = QQ[] quadratic = invariant_theory.ternary_quadratic(x^2+y^2+1 + t*x*y, [x,y]) quadratic.dual()
- classmethod from_invariants(discriminant, x, z, *args, **kwargs)[source]¶
Construct a binary quadratic from its discriminant.
This function constructs a binary quadratic whose discriminant equal the one provided as argument up to scaling.
INPUT:
discriminant– value of the discriminant used to reconstruct the binary quadratic
OUTPUT: a QuadraticForm with 2 variables
EXAMPLES:
sage: R.<x,y> = QQ[] sage: from sage.rings.invariants.invariant_theory import QuadraticForm sage: QuadraticForm.from_invariants(1, x, y) Binary quadratic with coefficients (1, -1/4, 0)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> from sage.rings.invariants.invariant_theory import QuadraticForm >>> QuadraticForm.from_invariants(Integer(1), x, y) Binary quadratic with coefficients (1, -1/4, 0)
R.<x,y> = QQ[] from sage.rings.invariants.invariant_theory import QuadraticForm QuadraticForm.from_invariants(1, x, y)
- invariants(type='discriminant')[source]¶
Return a tuple of invariants of a binary quadratic.
INPUT:
type– the type of invariants to return; the default choice is to return the discriminant
OUTPUT: the invariants of the binary quadratic
EXAMPLES:
sage: R.<x0, x1> = QQ[] sage: p = 2*x1^2 + 5*x0*x1 + 3*x0^2 sage: quadratic = invariant_theory.binary_quadratic(p, x0, x1) sage: quadratic.invariants() (1,) sage: quadratic.invariants('unknown') Traceback (most recent call last): ... ValueError: unknown type of invariants unknown for a binary quadratic
>>> from sage.all import * >>> R = QQ['x0, x1']; (x0, x1,) = R._first_ngens(2) >>> p = Integer(2)*x1**Integer(2) + Integer(5)*x0*x1 + Integer(3)*x0**Integer(2) >>> quadratic = invariant_theory.binary_quadratic(p, x0, x1) >>> quadratic.invariants() (1,) >>> quadratic.invariants('unknown') Traceback (most recent call last): ... ValueError: unknown type of invariants unknown for a binary quadratic
R.<x0, x1> = QQ[] p = 2*x1^2 + 5*x0*x1 + 3*x0^2 quadratic = invariant_theory.binary_quadratic(p, x0, x1) quadratic.invariants() quadratic.invariants('unknown')
- matrix()[source]¶
Return the quadratic form as a symmetric matrix.
OUTPUT:
This method returns a symmetric matrix \(A\) such that the quadratic \(Q\) equals
\[Q(x,y,z,\dots) = (x,y,\dots) A (x,y,\dots)^t\]EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2+x*y) sage: matrix(quadratic) [ 1 1/2 0] [1/2 1 0] [ 0 0 1] sage: quadratic._matrix_() == matrix(quadratic) True
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quadratic = invariant_theory.ternary_quadratic(x**Integer(2)+y**Integer(2)+z**Integer(2)+x*y) >>> matrix(quadratic) [ 1 1/2 0] [1/2 1 0] [ 0 0 1] >>> quadratic._matrix_() == matrix(quadratic) True
R.<x,y,z> = QQ[] quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2+x*y) matrix(quadratic) quadratic._matrix_() == matrix(quadratic)
- monomials()[source]¶
List the basis monomials in the form.
OUTPUT:
A tuple of monomials. They are in the same order as
coeffs().EXAMPLES:
sage: R.<x,y> = QQ[] sage: quadratic = invariant_theory.quadratic_form(x^2 + y^2) sage: quadratic.monomials() (x^2, y^2, x*y) sage: quadratic = invariant_theory.inhomogeneous_quadratic_form(x^2 + y^2) sage: quadratic.monomials() (x^2, y^2, 1, x*y, x, y)
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quadratic = invariant_theory.quadratic_form(x**Integer(2) + y**Integer(2)) >>> quadratic.monomials() (x^2, y^2, x*y) >>> quadratic = invariant_theory.inhomogeneous_quadratic_form(x**Integer(2) + y**Integer(2)) >>> quadratic.monomials() (x^2, y^2, 1, x*y, x, y)
R.<x,y> = QQ[] quadratic = invariant_theory.quadratic_form(x^2 + y^2) quadratic.monomials() quadratic = invariant_theory.inhomogeneous_quadratic_form(x^2 + y^2) quadratic.monomials()
- scaled_coeffs()[source]¶
The scaled coefficients of a quadratic form.
Given
\[f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} 2 a_{jk} x_j x_k\]this function returns \(a = (a_0, \cdots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)
EXAMPLES:
sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[] sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv Ternary quadratic with coefficients (a, b, c, d, e, f) sage: inv.coeffs() (a, b, c, d, e, f) sage: inv.scaled_coeffs() (a, b, c, 1/2*d, 1/2*e, 1/2*f)
>>> from sage.all import * >>> R = QQ['a, b, c, d, e, f, g, x, y, z']; (a, b, c, d, e, f, g, x, y, z,) = R._first_ngens(10) >>> p = a*x**Integer(2) + b*y**Integer(2) + c*z**Integer(2) + d*x*y + e*x*z + f*y*z >>> inv = invariant_theory.quadratic_form(p, x,y,z); inv Ternary quadratic with coefficients (a, b, c, d, e, f) >>> inv.coeffs() (a, b, c, d, e, f) >>> inv.scaled_coeffs() (a, b, c, 1/2*d, 1/2*e, 1/2*f)
R.<a,b,c,d,e,f,g, x,y,z> = QQ[] p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z inv = invariant_theory.quadratic_form(p, x,y,z); inv inv.coeffs() inv.scaled_coeffs()
- class sage.rings.invariants.invariant_theory.SeveralAlgebraicForms(forms)[source]¶
Bases:
FormsBaseThe base class of multiple algebraic forms (i.e. homogeneous polynomials).
You should only instantiate the derived classes of this base class.
See
AlgebraicFormfor the base class of a single algebraic form.INPUT:
forms– list/tuple/iterable of at least oneAlgebraicFormobject, all with the same number of variables. Interpreted as multiple homogeneous polynomials in a common polynomial ring.
EXAMPLES:
sage: from sage.rings.invariants.invariant_theory import AlgebraicForm, SeveralAlgebraicForms sage: R.<x,y> = QQ[] sage: p = AlgebraicForm(2, 2, x^2, (x,y)) sage: q = AlgebraicForm(2, 2, y^2, (x,y)) sage: pq = SeveralAlgebraicForms([p, q])
>>> from sage.all import * >>> from sage.rings.invariants.invariant_theory import AlgebraicForm, SeveralAlgebraicForms >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> p = AlgebraicForm(Integer(2), Integer(2), x**Integer(2), (x,y)) >>> q = AlgebraicForm(Integer(2), Integer(2), y**Integer(2), (x,y)) >>> pq = SeveralAlgebraicForms([p, q])
from sage.rings.invariants.invariant_theory import AlgebraicForm, SeveralAlgebraicForms R.<x,y> = QQ[] p = AlgebraicForm(2, 2, x^2, (x,y)) q = AlgebraicForm(2, 2, y^2, (x,y)) pq = SeveralAlgebraicForms([p, q])
- get_form(i)[source]¶
Return the \(i\)-th form.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: q1 = invariant_theory.quadratic_form(x^2 + y^2) sage: q2 = invariant_theory.quadratic_form(x*y) sage: from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms sage: q12 = SeveralAlgebraicForms([q1, q2]) sage: q12.get_form(0) is q1 True sage: q12.get_form(1) is q2 True sage: q12[0] is q12.get_form(0) # syntactic sugar True sage: q12[1] is q12.get_form(1) # syntactic sugar True
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> q1 = invariant_theory.quadratic_form(x**Integer(2) + y**Integer(2)) >>> q2 = invariant_theory.quadratic_form(x*y) >>> from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms >>> q12 = SeveralAlgebraicForms([q1, q2]) >>> q12.get_form(Integer(0)) is q1 True >>> q12.get_form(Integer(1)) is q2 True >>> q12[Integer(0)] is q12.get_form(Integer(0)) # syntactic sugar True >>> q12[Integer(1)] is q12.get_form(Integer(1)) # syntactic sugar True
R.<x,y> = QQ[] q1 = invariant_theory.quadratic_form(x^2 + y^2) q2 = invariant_theory.quadratic_form(x*y) from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms q12 = SeveralAlgebraicForms([q1, q2]) q12.get_form(0) is q1 q12.get_form(1) is q2 q12[0] is q12.get_form(0) # syntactic sugar q12[1] is q12.get_form(1) # syntactic sugar
- homogenized(var='h')[source]¶
Return form as defined by a homogeneous polynomial.
INPUT:
var– either a variable name, variable index or a variable (default:'h')
OUTPUT:
The same algebraic form, but defined by a homogeneous polynomial.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: q = invariant_theory.quaternary_biquadratic(x^2 + 1, y^2 + 1, [x,y,z]) sage: q Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0) sage: q.homogenized() Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0) sage: type(q) is type(q.homogenized()) True
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> q = invariant_theory.quaternary_biquadratic(x**Integer(2) + Integer(1), y**Integer(2) + Integer(1), [x,y,z]) >>> q Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0) >>> q.homogenized() Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0) and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0) >>> type(q) is type(q.homogenized()) True
R.<x,y,z> = QQ[] q = invariant_theory.quaternary_biquadratic(x^2 + 1, y^2 + 1, [x,y,z]) q q.homogenized() type(q) is type(q.homogenized())
- n_forms()[source]¶
Return the number of forms.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: q1 = invariant_theory.quadratic_form(x^2 + y^2) sage: q2 = invariant_theory.quadratic_form(x*y) sage: from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms sage: q12 = SeveralAlgebraicForms([q1, q2]) sage: q12.n_forms() 2 sage: len(q12) == q12.n_forms() # syntactic sugar True
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> q1 = invariant_theory.quadratic_form(x**Integer(2) + y**Integer(2)) >>> q2 = invariant_theory.quadratic_form(x*y) >>> from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms >>> q12 = SeveralAlgebraicForms([q1, q2]) >>> q12.n_forms() 2 >>> len(q12) == q12.n_forms() # syntactic sugar True
R.<x,y> = QQ[] q1 = invariant_theory.quadratic_form(x^2 + y^2) q2 = invariant_theory.quadratic_form(x*y) from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms q12 = SeveralAlgebraicForms([q1, q2]) q12.n_forms() len(q12) == q12.n_forms() # syntactic sugar
- class sage.rings.invariants.invariant_theory.TernaryCubic(n, d, polynomial, *args)[source]¶
Bases:
AlgebraicFormInvariant theory of a ternary cubic.
You should use the
invariant_theoryfactory object to construct instances of this class. Seeternary_cubic()for details.- Hessian()[source]¶
Return the Hessian covariant.
OUTPUT:
The Hessian matrix multiplied with the conventional normalization factor \(1/216\).
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) sage: cubic.Hessian() x*y*z sage: R.<x,y> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) sage: cubic.Hessian() x*y
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) >>> cubic.Hessian() x*y*z >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + Integer(1)) >>> cubic.Hessian() x*y
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) cubic.Hessian() R.<x,y> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) cubic.Hessian()
- J_covariant()[source]¶
Return the J-covariant of the ternary cubic.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) sage: cubic.J_covariant() x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6 sage: R.<x,y> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) sage: cubic.J_covariant() x^6*y^3 - x^3*y^6 - x^6 + y^6 + x^3 - y^3
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) >>> cubic.J_covariant() x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6 >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + Integer(1)) >>> cubic.J_covariant() x^6*y^3 - x^3*y^6 - x^6 + y^6 + x^3 - y^3
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) cubic.J_covariant() R.<x,y> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) cubic.J_covariant()
- S_invariant()[source]¶
Return the S-invariant.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^2*y + y^3 + z^3 + x*y*z) sage: cubic.S_invariant() -1/1296
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(2)*y + y**Integer(3) + z**Integer(3) + x*y*z) >>> cubic.S_invariant() -1/1296
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^2*y + y^3 + z^3 + x*y*z) cubic.S_invariant()
- T_invariant()[source]¶
Return the T-invariant.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) sage: cubic.T_invariant() 1 sage: R.<x,y,z,t> = GF(7)[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3 + t*x*y*z, [x,y,z]) sage: cubic.T_invariant() -t^6 - t^3 + 1
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) >>> cubic.T_invariant() 1 >>> R = GF(Integer(7))['x, y, z, t']; (x, y, z, t,) = R._first_ngens(4) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3) + t*x*y*z, [x,y,z]) >>> cubic.T_invariant() -t^6 - t^3 + 1
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) cubic.T_invariant() R.<x,y,z,t> = GF(7)[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3 + t*x*y*z, [x,y,z]) cubic.T_invariant()
- Theta_covariant()[source]¶
Return the \(\Theta\) covariant.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) sage: cubic.Theta_covariant() -x^3*y^3 - x^3*z^3 - y^3*z^3 sage: R.<x,y> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) sage: cubic.Theta_covariant() -x^3*y^3 - x^3 - y^3 sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + ....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) sage: cubic = invariant_theory.ternary_cubic(p, x,y,z) sage: len(list(cubic.Theta_covariant())) 6952
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + z**Integer(3)) >>> cubic.Theta_covariant() -x^3*y^3 - x^3*z^3 - y^3*z^3 >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3) + y**Integer(3) + Integer(1)) >>> cubic.Theta_covariant() -x^3*y^3 - x^3 - y^3 >>> R = QQ['x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00']; (x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00,) = R._first_ngens(13) >>> p = ( a30*x**Integer(3) + a21*x**Integer(2)*y + a12*x*y**Integer(2) + a03*y**Integer(3) + a20*x**Integer(2)*z + ... a11*x*y*z + a02*y**Integer(2)*z + a10*x*z**Integer(2) + a01*y*z**Integer(2) + a00*z**Integer(3) ) >>> cubic = invariant_theory.ternary_cubic(p, x,y,z) >>> len(list(cubic.Theta_covariant())) 6952
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + z^3) cubic.Theta_covariant() R.<x,y> = QQ[] cubic = invariant_theory.ternary_cubic(x^3 + y^3 + 1) cubic.Theta_covariant() R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) cubic = invariant_theory.ternary_cubic(p, x,y,z) len(list(cubic.Theta_covariant()))
- coeffs()[source]¶
Return the coefficients of a cubic.
Given
\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}, a_{20}, a_{12}, a_{02}, a_{10}, a_{01}, a_{11})\)
EXAMPLES:
sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + ....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) sage: invariant_theory.ternary_cubic(p, x,y,z).coeffs() (a30, a03, a00, a21, a20, a12, a02, a10, a01, a11) sage: invariant_theory.ternary_cubic(p.subs(z=1), x, y).coeffs() (a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)
>>> from sage.all import * >>> R = QQ['x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00']; (x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00,) = R._first_ngens(13) >>> p = ( a30*x**Integer(3) + a21*x**Integer(2)*y + a12*x*y**Integer(2) + a03*y**Integer(3) + a20*x**Integer(2)*z + ... a11*x*y*z + a02*y**Integer(2)*z + a10*x*z**Integer(2) + a01*y*z**Integer(2) + a00*z**Integer(3) ) >>> invariant_theory.ternary_cubic(p, x,y,z).coeffs() (a30, a03, a00, a21, a20, a12, a02, a10, a01, a11) >>> invariant_theory.ternary_cubic(p.subs(z=Integer(1)), x, y).coeffs() (a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)
R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) invariant_theory.ternary_cubic(p, x,y,z).coeffs() invariant_theory.ternary_cubic(p.subs(z=1), x, y).coeffs()
- monomials()[source]¶
List the basis monomials of the form.
OUTPUT:
A tuple of monomials. They are in the same order as
coeffs().EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: cubic = invariant_theory.ternary_cubic(x^3+y*z^2) sage: cubic.monomials() (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, y^2*z, x*z^2, y*z^2, x*y*z)
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> cubic = invariant_theory.ternary_cubic(x**Integer(3)+y*z**Integer(2)) >>> cubic.monomials() (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, y^2*z, x*z^2, y*z^2, x*y*z)
R.<x,y,z> = QQ[] cubic = invariant_theory.ternary_cubic(x^3+y*z^2) cubic.monomials()
- polar_conic()[source]¶
Return the polar conic of the cubic.
OUTPUT:
Given the ternary cubic \(f(X,Y,Z)\), this method returns the symmetric matrix \(A(x,y,z)\) defined by
\[x f_X + y f_Y + z f_Z = (X,Y,Z) \cdot A(x,y,z) \cdot (X,Y,Z)^t\]EXAMPLES:
sage: R.<x,y,z,X,Y,Z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + ....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) sage: cubic = invariant_theory.ternary_cubic(p, x,y,z) sage: cubic.polar_conic() [ 3*x*a30 + y*a21 + z*a20 x*a21 + y*a12 + 1/2*z*a11 x*a20 + 1/2*y*a11 + z*a10] [x*a21 + y*a12 + 1/2*z*a11 x*a12 + 3*y*a03 + z*a02 1/2*x*a11 + y*a02 + z*a01] [x*a20 + 1/2*y*a11 + z*a10 1/2*x*a11 + y*a02 + z*a01 x*a10 + y*a01 + 3*z*a00] sage: polar_eqn = X*p.derivative(x) + Y*p.derivative(y) + Z*p.derivative(z) sage: polar = invariant_theory.ternary_quadratic(polar_eqn, [x,y,z]) sage: polar.matrix().subs(X=x,Y=y,Z=z) == cubic.polar_conic() True
>>> from sage.all import * >>> R = QQ['x, y, z, X, Y, Z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00']; (x, y, z, X, Y, Z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00,) = R._first_ngens(16) >>> p = ( a30*x**Integer(3) + a21*x**Integer(2)*y + a12*x*y**Integer(2) + a03*y**Integer(3) + a20*x**Integer(2)*z + ... a11*x*y*z + a02*y**Integer(2)*z + a10*x*z**Integer(2) + a01*y*z**Integer(2) + a00*z**Integer(3) ) >>> cubic = invariant_theory.ternary_cubic(p, x,y,z) >>> cubic.polar_conic() [ 3*x*a30 + y*a21 + z*a20 x*a21 + y*a12 + 1/2*z*a11 x*a20 + 1/2*y*a11 + z*a10] [x*a21 + y*a12 + 1/2*z*a11 x*a12 + 3*y*a03 + z*a02 1/2*x*a11 + y*a02 + z*a01] [x*a20 + 1/2*y*a11 + z*a10 1/2*x*a11 + y*a02 + z*a01 x*a10 + y*a01 + 3*z*a00] >>> polar_eqn = X*p.derivative(x) + Y*p.derivative(y) + Z*p.derivative(z) >>> polar = invariant_theory.ternary_quadratic(polar_eqn, [x,y,z]) >>> polar.matrix().subs(X=x,Y=y,Z=z) == cubic.polar_conic() True
R.<x,y,z,X,Y,Z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) cubic = invariant_theory.ternary_cubic(p, x,y,z) cubic.polar_conic() polar_eqn = X*p.derivative(x) + Y*p.derivative(y) + Z*p.derivative(z) polar = invariant_theory.ternary_quadratic(polar_eqn, [x,y,z]) polar.matrix().subs(X=x,Y=y,Z=z) == cubic.polar_conic()
- scaled_coeffs()[source]¶
Return the coefficients of a cubic.
Compared to
coeffs(), this method returns rescaled coefficients that are often used in invariant theory.Given
\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}/3, a_{20}/3, a_{12}/3, a_{02}/3, a_{10}/3, a_{01}/3, a_{11}/6)\)
EXAMPLES:
sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + ....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) sage: invariant_theory.ternary_cubic(p, x,y,z).scaled_coeffs() (a30, a03, a00, 1/3*a21, 1/3*a20, 1/3*a12, 1/3*a02, 1/3*a10, 1/3*a01, 1/6*a11)
>>> from sage.all import * >>> R = QQ['x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00']; (x, y, z, a30, a21, a12, a03, a20, a11, a02, a10, a01, a00,) = R._first_ngens(13) >>> p = ( a30*x**Integer(3) + a21*x**Integer(2)*y + a12*x*y**Integer(2) + a03*y**Integer(3) + a20*x**Integer(2)*z + ... a11*x*y*z + a02*y**Integer(2)*z + a10*x*z**Integer(2) + a01*y*z**Integer(2) + a00*z**Integer(3) ) >>> invariant_theory.ternary_cubic(p, x,y,z).scaled_coeffs() (a30, a03, a00, 1/3*a21, 1/3*a20, 1/3*a12, 1/3*a02, 1/3*a10, 1/3*a01, 1/6*a11)
R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z + a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 ) invariant_theory.ternary_cubic(p, x,y,z).scaled_coeffs()
- syzygy(U, S, T, H, Theta, J)[source]¶
Return the syzygy of the cubic evaluated on the invariants and covariants.
INPUT:
U,S,T,H,Theta,J– polynomials from the same polynomial ring.
OUTPUT:
0 if evaluated for the form, the S invariant, the T invariant, the Hessian, the \(\Theta\) covariant and the J-covariant of a ternary cubic.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: monomials = (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, ....: y^2*z, x*z^2, y*z^2, x*y*z) sage: random_poly = sum([ randint(0,10000) * m for m in monomials ]) sage: cubic = invariant_theory.ternary_cubic(random_poly) sage: U = cubic.form() sage: S = cubic.S_invariant() sage: T = cubic.T_invariant() sage: H = cubic.Hessian() sage: Theta = cubic.Theta_covariant() sage: J = cubic.J_covariant() sage: cubic.syzygy(U, S, T, H, Theta, J) 0
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> monomials = (x**Integer(3), y**Integer(3), z**Integer(3), x**Integer(2)*y, x**Integer(2)*z, x*y**Integer(2), ... y**Integer(2)*z, x*z**Integer(2), y*z**Integer(2), x*y*z) >>> random_poly = sum([ randint(Integer(0),Integer(10000)) * m for m in monomials ]) >>> cubic = invariant_theory.ternary_cubic(random_poly) >>> U = cubic.form() >>> S = cubic.S_invariant() >>> T = cubic.T_invariant() >>> H = cubic.Hessian() >>> Theta = cubic.Theta_covariant() >>> J = cubic.J_covariant() >>> cubic.syzygy(U, S, T, H, Theta, J) 0
R.<x,y,z> = QQ[] monomials = (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, y^2*z, x*z^2, y*z^2, x*y*z) random_poly = sum([ randint(0,10000) * m for m in monomials ]) cubic = invariant_theory.ternary_cubic(random_poly) U = cubic.form() S = cubic.S_invariant() T = cubic.T_invariant() H = cubic.Hessian() Theta = cubic.Theta_covariant() J = cubic.J_covariant() cubic.syzygy(U, S, T, H, Theta, J)
- class sage.rings.invariants.invariant_theory.TernaryQuadratic(n, d, polynomial, *args)[source]¶
Bases:
QuadraticFormInvariant theory of a ternary quadratic.
You should use the
invariant_theoryfactory object to construct instances of this class. Seeternary_quadratic()for details.- coeffs()[source]¶
Return the coefficients of a quadratic.
Given
\[p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\]this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}, a_{10}, a_{01} )\)
EXAMPLES:
sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a20*x^2 + a11*x*y + a02*y^2 + ....: a10*x*z + a01*y*z + a00*z^2 ) sage: invariant_theory.ternary_quadratic(p, x,y,z).coeffs() (a20, a02, a00, a11, a10, a01) sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).coeffs() (a20, a02, a00, a11, a10, a01)
>>> from sage.all import * >>> R = QQ['x, y, z, a20, a11, a02, a10, a01, a00']; (x, y, z, a20, a11, a02, a10, a01, a00,) = R._first_ngens(9) >>> p = ( a20*x**Integer(2) + a11*x*y + a02*y**Integer(2) + ... a10*x*z + a01*y*z + a00*z**Integer(2) ) >>> invariant_theory.ternary_quadratic(p, x,y,z).coeffs() (a20, a02, a00, a11, a10, a01) >>> invariant_theory.ternary_quadratic(p.subs(z=Integer(1)), x, y).coeffs() (a20, a02, a00, a11, a10, a01)
R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a20*x^2 + a11*x*y + a02*y^2 + a10*x*z + a01*y*z + a00*z^2 ) invariant_theory.ternary_quadratic(p, x,y,z).coeffs() invariant_theory.ternary_quadratic(p.subs(z=1), x, y).coeffs()
- covariant_conic(other)[source]¶
Return the ternary quadratic covariant to
selfandother.INPUT:
other– another ternary quadratic
OUTPUT:
The so-called covariant conic, a ternary quadratic. It is symmetric under exchange of
selfandother.EXAMPLES:
sage: ring.<x,y,z> = QQ[] sage: Q = invariant_theory.ternary_quadratic(x^2 + y^2 + z^2) sage: R = invariant_theory.ternary_quadratic(x*y + x*z + y*z) sage: Q.covariant_conic(R) -x*y - x*z - y*z sage: R.covariant_conic(Q) -x*y - x*z - y*z
>>> from sage.all import * >>> ring = QQ['x, y, z']; (x, y, z,) = ring._first_ngens(3) >>> Q = invariant_theory.ternary_quadratic(x**Integer(2) + y**Integer(2) + z**Integer(2)) >>> R = invariant_theory.ternary_quadratic(x*y + x*z + y*z) >>> Q.covariant_conic(R) -x*y - x*z - y*z >>> R.covariant_conic(Q) -x*y - x*z - y*z
ring.<x,y,z> = QQ[] Q = invariant_theory.ternary_quadratic(x^2 + y^2 + z^2) R = invariant_theory.ternary_quadratic(x*y + x*z + y*z) Q.covariant_conic(R) R.covariant_conic(Q)
- monomials()[source]¶
List the basis monomials of the form.
OUTPUT:
A tuple of monomials. They are in the same order as
coeffs().EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: quadratic = invariant_theory.ternary_quadratic(x^2 + y*z) sage: quadratic.monomials() (x^2, y^2, z^2, x*y, x*z, y*z)
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> quadratic = invariant_theory.ternary_quadratic(x**Integer(2) + y*z) >>> quadratic.monomials() (x^2, y^2, z^2, x*y, x*z, y*z)
R.<x,y,z> = QQ[] quadratic = invariant_theory.ternary_quadratic(x^2 + y*z) quadratic.monomials()
- scaled_coeffs()[source]¶
Return the scaled coefficients of a quadratic.
Given
\[p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\]this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}/2, a_{10}/2, a_{01}/2, )\)
EXAMPLES:
sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[] sage: p = ( a20*x^2 + a11*x*y + a02*y^2 + ....: a10*x*z + a01*y*z + a00*z^2 ) sage: invariant_theory.ternary_quadratic(p, x,y,z).scaled_coeffs() (a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01) sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).scaled_coeffs() (a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)
>>> from sage.all import * >>> R = QQ['x, y, z, a20, a11, a02, a10, a01, a00']; (x, y, z, a20, a11, a02, a10, a01, a00,) = R._first_ngens(9) >>> p = ( a20*x**Integer(2) + a11*x*y + a02*y**Integer(2) + ... a10*x*z + a01*y*z + a00*z**Integer(2) ) >>> invariant_theory.ternary_quadratic(p, x,y,z).scaled_coeffs() (a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01) >>> invariant_theory.ternary_quadratic(p.subs(z=Integer(1)), x, y).scaled_coeffs() (a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)
R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[] p = ( a20*x^2 + a11*x*y + a02*y^2 + a10*x*z + a01*y*z + a00*z^2 ) invariant_theory.ternary_quadratic(p, x,y,z).scaled_coeffs() invariant_theory.ternary_quadratic(p.subs(z=1), x, y).scaled_coeffs()
- class sage.rings.invariants.invariant_theory.TwoAlgebraicForms(forms)[source]¶
Bases:
SeveralAlgebraicForms- first()[source]¶
Return the first of the two forms.
OUTPUT: the first algebraic form used in the definition
EXAMPLES:
sage: R.<x,y> = QQ[] sage: q0 = invariant_theory.quadratic_form(x^2 + y^2) sage: q1 = invariant_theory.quadratic_form(x*y) sage: from sage.rings.invariants.invariant_theory import TwoAlgebraicForms sage: q = TwoAlgebraicForms([q0, q1]) sage: q.first() is q0 True sage: q.get_form(0) is q0 True sage: q.first().polynomial() x^2 + y^2
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> q0 = invariant_theory.quadratic_form(x**Integer(2) + y**Integer(2)) >>> q1 = invariant_theory.quadratic_form(x*y) >>> from sage.rings.invariants.invariant_theory import TwoAlgebraicForms >>> q = TwoAlgebraicForms([q0, q1]) >>> q.first() is q0 True >>> q.get_form(Integer(0)) is q0 True >>> q.first().polynomial() x^2 + y^2
R.<x,y> = QQ[] q0 = invariant_theory.quadratic_form(x^2 + y^2) q1 = invariant_theory.quadratic_form(x*y) from sage.rings.invariants.invariant_theory import TwoAlgebraicForms q = TwoAlgebraicForms([q0, q1]) q.first() is q0 q.get_form(0) is q0 q.first().polynomial()
- second()[source]¶
Return the second of the two forms.
OUTPUT: the second form used in the definition
EXAMPLES:
sage: R.<x,y> = QQ[] sage: q0 = invariant_theory.quadratic_form(x^2 + y^2) sage: q1 = invariant_theory.quadratic_form(x*y) sage: from sage.rings.invariants.invariant_theory import TwoAlgebraicForms sage: q = TwoAlgebraicForms([q0, q1]) sage: q.second() is q1 True sage: q.get_form(1) is q1 True sage: q.second().polynomial() x*y
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> q0 = invariant_theory.quadratic_form(x**Integer(2) + y**Integer(2)) >>> q1 = invariant_theory.quadratic_form(x*y) >>> from sage.rings.invariants.invariant_theory import TwoAlgebraicForms >>> q = TwoAlgebraicForms([q0, q1]) >>> q.second() is q1 True >>> q.get_form(Integer(1)) is q1 True >>> q.second().polynomial() x*y
R.<x,y> = QQ[] q0 = invariant_theory.quadratic_form(x^2 + y^2) q1 = invariant_theory.quadratic_form(x*y) from sage.rings.invariants.invariant_theory import TwoAlgebraicForms q = TwoAlgebraicForms([q0, q1]) q.second() is q1 q.get_form(1) is q1 q.second().polynomial()
- class sage.rings.invariants.invariant_theory.TwoQuaternaryQuadratics(forms)[source]¶
Bases:
TwoAlgebraicFormsInvariant theory of two quaternary quadratics.
You should use the
invariant_theoryfactory object to construct instances of this class. Seequaternary_biquadratics()for details.REFERENCES:
- Delta_invariant()[source]¶
Return the \(\Delta\) invariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Delta_invariant() == coeffs[4] True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Delta_invariant() == coeffs[Integer(4)] True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Delta_invariant() == coeffs[4]
- Delta_prime_invariant()[source]¶
Return the \(\Delta'\) invariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Delta_prime_invariant() == coeffs[0] True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Delta_prime_invariant() == coeffs[Integer(0)] True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Delta_prime_invariant() == coeffs[0]
- J_covariant()[source]¶
The \(J\)-covariant.
This is the Jacobian determinant of the two biquadratics, the \(T\)-covariant, and the \(T'\)-covariant with respect to the four homogeneous variables.
EXAMPLES:
sage: R.<w,x,y,z,a0,a1,a2,a3,A0,A1,A2,A3> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3*w^2 sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3*w^2 sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [w, x, y, z]) sage: q.J_covariant().factor() z * y * x * w * (a3*A2 - a2*A3) * (a3*A1 - a1*A3) * (-a2*A1 + a1*A2) * (a3*A0 - a0*A3) * (-a2*A0 + a0*A2) * (-a1*A0 + a0*A1)
>>> from sage.all import * >>> R = QQ['w, x, y, z, a0, a1, a2, a3, A0, A1, A2, A3']; (w, x, y, z, a0, a1, a2, a3, A0, A1, A2, A3,) = R._first_ngens(12) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3*w**Integer(2) >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3*w**Integer(2) >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [w, x, y, z]) >>> q.J_covariant().factor() z * y * x * w * (a3*A2 - a2*A3) * (a3*A1 - a1*A3) * (-a2*A1 + a1*A2) * (a3*A0 - a0*A3) * (-a2*A0 + a0*A2) * (-a1*A0 + a0*A1)
R.<w,x,y,z,a0,a1,a2,a3,A0,A1,A2,A3> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3*w^2 p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3*w^2 q = invariant_theory.quaternary_biquadratic(p1, p2, [w, x, y, z]) q.J_covariant().factor()
- Phi_invariant()[source]¶
Return the \(\Phi'\) invariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Phi_invariant() == coeffs[2] True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Phi_invariant() == coeffs[Integer(2)] True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Phi_invariant() == coeffs[2]
- T_covariant()[source]¶
The \(T\)-covariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: T = invariant_theory.quaternary_quadratic(q.T_covariant(), [x,y,z]).matrix() sage: M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate() sage: M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) ....: lambda m: m.coefficient(t)) sage: M == q.Delta_invariant()*T # long time True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> T = invariant_theory.quaternary_quadratic(q.T_covariant(), [x,y,z]).matrix() >>> M = q[Integer(0)].matrix().adjugate() + t*q[Integer(1)].matrix().adjugate() >>> M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) ... lambda m: m.coefficient(t)) >>> M == q.Delta_invariant()*T # long time True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) T = invariant_theory.quaternary_quadratic(q.T_covariant(), [x,y,z]).matrix() M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate() M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) lambda m: m.coefficient(t)) M == q.Delta_invariant()*T # long time
- T_prime_covariant()[source]¶
The \(T'\)-covariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: Tprime = invariant_theory.quaternary_quadratic( ....: q.T_prime_covariant(), [x,y,z]).matrix() sage: M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate() sage: M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) ....: lambda m: m.coefficient(t^2)) sage: M == q.Delta_prime_invariant() * Tprime # long time True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> Tprime = invariant_theory.quaternary_quadratic( ... q.T_prime_covariant(), [x,y,z]).matrix() >>> M = q[Integer(0)].matrix().adjugate() + t*q[Integer(1)].matrix().adjugate() >>> M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) ... lambda m: m.coefficient(t**Integer(2))) >>> M == q.Delta_prime_invariant() * Tprime # long time True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) Tprime = invariant_theory.quaternary_quadratic( q.T_prime_covariant(), [x,y,z]).matrix() M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate() M = M.adjugate().apply_map( # long time (4s on my thinkpad W530) lambda m: m.coefficient(t^2)) M == q.Delta_prime_invariant() * Tprime # long time
- Theta_invariant()[source]¶
Return the \(\Theta\) invariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Theta_invariant() == coeffs[3] True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Theta_invariant() == coeffs[Integer(3)] True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Theta_invariant() == coeffs[3]
- Theta_prime_invariant()[source]¶
Return the \(\Theta'\) invariant.
EXAMPLES:
sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Theta_prime_invariant() == coeffs[1] True
>>> from sage.all import * >>> R = QQ['x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5']; (x, y, z, t, a0, a1, a2, a3, b0, b1, b2, b3, b4, b5, A0, A1, A2, A3, B0, B1, B2, B3, B4, B5,) = R._first_ngens(24) >>> p1 = a0*x**Integer(2) + a1*y**Integer(2) + a2*z**Integer(2) + a3 >>> p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z >>> p2 = A0*x**Integer(2) + A1*y**Integer(2) + A2*z**Integer(2) + A3 >>> p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z >>> q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Theta_prime_invariant() == coeffs[Integer(1)] True
R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[] p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3 p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3 p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Theta_prime_invariant() == coeffs[1]
- syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)[source]¶
Return the syzygy evaluated on the invariants and covariants.
INPUT:
Delta,Theta,Phi,Theta_prime,Delta_prime,U,V,T,T_prime,J– polynomials from the same polynomial ring.
OUTPUT:
Zero if the
Uis the first polynomial,Vthe second polynomial, and the remaining input are the invariants and covariants of a quaternary biquadratic.EXAMPLES:
sage: R.<w,x,y,z> = QQ[] sage: monomials = [x^2, x*y, y^2, x*z, y*z, z^2, x*w, y*w, z*w, w^2] sage: def q_rnd(): return sum(randint(-1000, 1000)*m for m in monomials) sage: biquadratic = invariant_theory.quaternary_biquadratic(q_rnd(), q_rnd()) sage: Delta = biquadratic.Delta_invariant() sage: Theta = biquadratic.Theta_invariant() sage: Phi = biquadratic.Phi_invariant() sage: Theta_prime = biquadratic.Theta_prime_invariant() sage: Delta_prime = biquadratic.Delta_prime_invariant() sage: U = biquadratic.first().polynomial() sage: V = biquadratic.second().polynomial() sage: T = biquadratic.T_covariant() sage: T_prime = biquadratic.T_prime_covariant() sage: J = biquadratic.J_covariant() sage: biquadratic.syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J) 0
>>> from sage.all import * >>> R = QQ['w, x, y, z']; (w, x, y, z,) = R._first_ngens(4) >>> monomials = [x**Integer(2), x*y, y**Integer(2), x*z, y*z, z**Integer(2), x*w, y*w, z*w, w**Integer(2)] >>> def q_rnd(): return sum(randint(-Integer(1000), Integer(1000))*m for m in monomials) >>> biquadratic = invariant_theory.quaternary_biquadratic(q_rnd(), q_rnd()) >>> Delta = biquadratic.Delta_invariant() >>> Theta = biquadratic.Theta_invariant() >>> Phi = biquadratic.Phi_invariant() >>> Theta_prime = biquadratic.Theta_prime_invariant() >>> Delta_prime = biquadratic.Delta_prime_invariant() >>> U = biquadratic.first().polynomial() >>> V = biquadratic.second().polynomial() >>> T = biquadratic.T_covariant() >>> T_prime = biquadratic.T_prime_covariant() >>> J = biquadratic.J_covariant() >>> biquadratic.syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J) 0
R.<w,x,y,z> = QQ[] monomials = [x^2, x*y, y^2, x*z, y*z, z^2, x*w, y*w, z*w, w^2] def q_rnd(): return sum(randint(-1000, 1000)*m for m in monomials) biquadratic = invariant_theory.quaternary_biquadratic(q_rnd(), q_rnd()) Delta = biquadratic.Delta_invariant() Theta = biquadratic.Theta_invariant() Phi = biquadratic.Phi_invariant() Theta_prime = biquadratic.Theta_prime_invariant() Delta_prime = biquadratic.Delta_prime_invariant() U = biquadratic.first().polynomial() V = biquadratic.second().polynomial() T = biquadratic.T_covariant() T_prime = biquadratic.T_prime_covariant() J = biquadratic.J_covariant() biquadratic.syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)
If the arguments are not the invariants and covariants then the output is some (generically nonzero) polynomial:
sage: biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, 1, 1, x) -x^2 + 1
>>> from sage.all import * >>> biquadratic.syzygy(Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), x) -x^2 + 1
biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, 1, 1, x)
- class sage.rings.invariants.invariant_theory.TwoTernaryQuadratics(forms)[source]¶
Bases:
TwoAlgebraicFormsInvariant theory of two ternary quadratics.
You should use the
invariant_theoryfactory object to construct instances of this class. Seeternary_biquadratics()for details.REFERENCES:
Section on “Invariants and Covariants of Systems of Conics”, Art. 388 (a) in [Sal1954]
- Delta_invariant()[source]¶
Return the \(\Delta\) invariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Delta_invariant() == coeffs[3] True
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t,) = R._first_ngens(16) >>> p1 = a00*y0**Integer(2) + Integer(2)*a01*y0*y1 + a11*y1**Integer(2) + Integer(2)*a02*y0*y2 + Integer(2)*a12*y1*y2 + a22*y2**Integer(2) >>> p2 = b00*y0**Integer(2) + Integer(2)*b01*y0*y1 + b11*y1**Integer(2) + Integer(2)*b02*y0*y2 + Integer(2)*b12*y1*y2 + b22*y2**Integer(2) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Delta_invariant() == coeffs[Integer(3)] True
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Delta_invariant() == coeffs[3]
- Delta_prime_invariant()[source]¶
Return the \(\Delta'\) invariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Delta_prime_invariant() == coeffs[0] True
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t,) = R._first_ngens(16) >>> p1 = a00*y0**Integer(2) + Integer(2)*a01*y0*y1 + a11*y1**Integer(2) + Integer(2)*a02*y0*y2 + Integer(2)*a12*y1*y2 + a22*y2**Integer(2) >>> p2 = b00*y0**Integer(2) + Integer(2)*b01*y0*y1 + b11*y1**Integer(2) + Integer(2)*b02*y0*y2 + Integer(2)*b12*y1*y2 + b22*y2**Integer(2) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Delta_prime_invariant() == coeffs[Integer(0)] True
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Delta_prime_invariant() == coeffs[0]
- F_covariant()[source]¶
Return the \(F\) covariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[] sage: p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57 sage: p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) sage: q.F_covariant() -32566577*x^2 + 29060637/2*x*y + 20153633/4*y^2 - 30250497/2*x - 241241273/4*y - 323820473/16
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y,) = R._first_ngens(14) >>> p1 = Integer(73)*x**Integer(2) + Integer(96)*x*y - Integer(11)*y**Integer(2) + Integer(4)*x + Integer(63)*y + Integer(57) >>> p2 = Integer(61)*x**Integer(2) - Integer(100)*x*y - Integer(72)*y**Integer(2) - Integer(81)*x + Integer(39)*y - Integer(7) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) >>> q.F_covariant() -32566577*x^2 + 29060637/2*x*y + 20153633/4*y^2 - 30250497/2*x - 241241273/4*y - 323820473/16
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[] p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57 p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7 q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) q.F_covariant()
- J_covariant()[source]¶
Return the \(J\) covariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[] sage: p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57 sage: p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) sage: q.J_covariant() 1057324024445*x^3 + 1209531088209*x^2*y + 942116599708*x*y^2 + 984553030871*y^3 + 543715345505/2*x^2 - 3065093506021/2*x*y + 755263948570*y^2 - 1118430692650*x - 509948695327/4*y + 3369951531745/8
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y,) = R._first_ngens(14) >>> p1 = Integer(73)*x**Integer(2) + Integer(96)*x*y - Integer(11)*y**Integer(2) + Integer(4)*x + Integer(63)*y + Integer(57) >>> p2 = Integer(61)*x**Integer(2) - Integer(100)*x*y - Integer(72)*y**Integer(2) - Integer(81)*x + Integer(39)*y - Integer(7) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) >>> q.J_covariant() 1057324024445*x^3 + 1209531088209*x^2*y + 942116599708*x*y^2 + 984553030871*y^3 + 543715345505/2*x^2 - 3065093506021/2*x*y + 755263948570*y^2 - 1118430692650*x - 509948695327/4*y + 3369951531745/8
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[] p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57 p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7 q = invariant_theory.ternary_biquadratic(p1, p2, [x, y]) q.J_covariant()
- Theta_invariant()[source]¶
Return the \(\Theta\) invariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Theta_invariant() == coeffs[2] True
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t,) = R._first_ngens(16) >>> p1 = a00*y0**Integer(2) + Integer(2)*a01*y0*y1 + a11*y1**Integer(2) + Integer(2)*a02*y0*y2 + Integer(2)*a12*y1*y2 + a22*y2**Integer(2) >>> p2 = b00*y0**Integer(2) + Integer(2)*b01*y0*y1 + b11*y1**Integer(2) + Integer(2)*b02*y0*y2 + Integer(2)*b12*y1*y2 + b22*y2**Integer(2) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Theta_invariant() == coeffs[Integer(2)] True
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Theta_invariant() == coeffs[2]
- Theta_prime_invariant()[source]¶
Return the \(\Theta'\) invariant.
EXAMPLES:
sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) sage: q.Theta_prime_invariant() == coeffs[1] True
>>> from sage.all import * >>> R = QQ['a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t']; (a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t,) = R._first_ngens(16) >>> p1 = a00*y0**Integer(2) + Integer(2)*a01*y0*y1 + a11*y1**Integer(2) + Integer(2)*a02*y0*y2 + Integer(2)*a12*y1*y2 + a22*y2**Integer(2) >>> p2 = b00*y0**Integer(2) + Integer(2)*b01*y0*y1 + b11*y1**Integer(2) + Integer(2)*b02*y0*y2 + Integer(2)*b12*y1*y2 + b22*y2**Integer(2) >>> q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) >>> coeffs = det(t * q[Integer(0)].matrix() + q[Integer(1)].matrix()).polynomial(t).coefficients(sparse=False) >>> q.Theta_prime_invariant() == coeffs[Integer(1)] True
R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[] p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2 p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2 q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2]) coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False) q.Theta_prime_invariant() == coeffs[1]
- syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J)[source]¶
Return the syzygy evaluated on the invariants and covariants.
INPUT:
Delta,Theta,Theta_prime,Delta_prime,S,S_prime,F,J– polynomials from the same polynomial ring.
OUTPUT:
Zero if
Sis the first polynomial,S_primethe second polynomial, and the remaining input are the invariants and covariants of a ternary biquadratic.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: monomials = [x^2, x*y, y^2, x*z, y*z, z^2] sage: def q_rnd(): return sum(randint(-1000, 1000)*m for m in monomials) sage: biquadratic = invariant_theory.ternary_biquadratic(q_rnd(), q_rnd(), [x,y,z]) sage: Delta = biquadratic.Delta_invariant() sage: Theta = biquadratic.Theta_invariant() sage: Theta_prime = biquadratic.Theta_prime_invariant() sage: Delta_prime = biquadratic.Delta_prime_invariant() sage: S = biquadratic.first().polynomial() sage: S_prime = biquadratic.second().polynomial() sage: F = biquadratic.F_covariant() sage: J = biquadratic.J_covariant() sage: biquadratic.syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J) 0
>>> from sage.all import * >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> monomials = [x**Integer(2), x*y, y**Integer(2), x*z, y*z, z**Integer(2)] >>> def q_rnd(): return sum(randint(-Integer(1000), Integer(1000))*m for m in monomials) >>> biquadratic = invariant_theory.ternary_biquadratic(q_rnd(), q_rnd(), [x,y,z]) >>> Delta = biquadratic.Delta_invariant() >>> Theta = biquadratic.Theta_invariant() >>> Theta_prime = biquadratic.Theta_prime_invariant() >>> Delta_prime = biquadratic.Delta_prime_invariant() >>> S = biquadratic.first().polynomial() >>> S_prime = biquadratic.second().polynomial() >>> F = biquadratic.F_covariant() >>> J = biquadratic.J_covariant() >>> biquadratic.syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J) 0
R.<x,y,z> = QQ[] monomials = [x^2, x*y, y^2, x*z, y*z, z^2] def q_rnd(): return sum(randint(-1000, 1000)*m for m in monomials) biquadratic = invariant_theory.ternary_biquadratic(q_rnd(), q_rnd(), [x,y,z]) Delta = biquadratic.Delta_invariant() Theta = biquadratic.Theta_invariant() Theta_prime = biquadratic.Theta_prime_invariant() Delta_prime = biquadratic.Delta_prime_invariant() S = biquadratic.first().polynomial() S_prime = biquadratic.second().polynomial() F = biquadratic.F_covariant() J = biquadratic.J_covariant() biquadratic.syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J)
If the arguments are not the invariants and covariants then the output is some (generically nonzero) polynomial:
sage: biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, x) 1/64*x^2 + 1
>>> from sage.all import * >>> biquadratic.syzygy(Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), Integer(1), x) 1/64*x^2 + 1
biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, x)
- sage.rings.invariants.invariant_theory.transvectant(f, g, h=1, scale='default')[source]¶
Return the h-th transvectant of f and g.
INPUT:
f,g– two homogeneous binary forms in the same polynomial ringh– the order of the transvectant; if it is not specified, the first transvectant is returnedscale– the scaling factor applied to the result. Possible values are'default'and'none'. The'default'scaling factor is the one that appears in the output statement below, if the scaling factor is'none'the quotient of factorials is left out.
OUTPUT:
The h-th transvectant of the listed forms \(f\) and \(g\):
\[(f,g)_h = \frac{(d_f-h)! \cdot (d_g-h)!}{d_f! \cdot d_g!}\left( \left(\frac{\partial}{\partial x}\frac{\partial}{\partial z'} - \frac{\partial}{\partial x'}\frac{\partial}{\partial z} \right)^h \left(f(x,z) \cdot g(x',z')\right) \right)_{(x',z')=(x,z)}\]EXAMPLES:
sage: from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant sage: R.<x,y> = QQ[] sage: f = AlgebraicForm(2, 5, x^5 + 5*x^4*y + 5*x*y^4 + y^5) sage: transvectant(f, f, 4) Binary quadratic given by 2*x^2 - 4*x*y + 2*y^2 sage: transvectant(f, f, 8) Binary form of degree -6 given by 0
>>> from sage.all import * >>> from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> f = AlgebraicForm(Integer(2), Integer(5), x**Integer(5) + Integer(5)*x**Integer(4)*y + Integer(5)*x*y**Integer(4) + y**Integer(5)) >>> transvectant(f, f, Integer(4)) Binary quadratic given by 2*x^2 - 4*x*y + 2*y^2 >>> transvectant(f, f, Integer(8)) Binary form of degree -6 given by 0
from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant R.<x,y> = QQ[] f = AlgebraicForm(2, 5, x^5 + 5*x^4*y + 5*x*y^4 + y^5) transvectant(f, f, 4) transvectant(f, f, 8)
The default scaling will yield an error for fields of positive characteristic below \(d_f!\) or \(d_g!\) as the denominator of the scaling factor will not be invertible in that case. The scale argument
'none'can be used to compute the transvectant in this case:sage: # needs sage.rings.finite_rings sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = GF(5)[] sage: f = AlgebraicForm(2, 5, a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 ....: + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5, x0, x1) sage: transvectant(f, f, 4) Traceback (most recent call last): ... ZeroDivisionError sage: transvectant(f, f, 4, scale='none') Binary quadratic given by -a3^2*x0^2 + a2*a4*x0^2 + a2*a3*x0*x1 - a1*a4*x0*x1 - a2^2*x1^2 + a1*a3*x1^2
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> R = GF(Integer(5))['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> f = AlgebraicForm(Integer(2), Integer(5), a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) ... + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5), x0, x1) >>> transvectant(f, f, Integer(4)) Traceback (most recent call last): ... ZeroDivisionError >>> transvectant(f, f, Integer(4), scale='none') Binary quadratic given by -a3^2*x0^2 + a2*a4*x0^2 + a2*a3*x0*x1 - a1*a4*x0*x1 - a2^2*x1^2 + a1*a3*x1^2
# needs sage.rings.finite_rings R.<a0,a1,a2,a3,a4,a5,x0,x1> = GF(5)[] f = AlgebraicForm(2, 5, a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5, x0, x1) transvectant(f, f, 4) transvectant(f, f, 4, scale='none')The additional factors that appear when
scale='none'is used can be seen if we consider the same transvectant over the rationals and compare it to the scaled version:sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[] sage: f = AlgebraicForm(2, 5, a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 ....: + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5, x0, x1) sage: transvectant(f, f, 4) Binary quadratic given by 3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2 + 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1 - 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1 + 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2 + 2/5*a0*a4*x1^2 sage: transvectant(f, f, 4, scale='none') Binary quadratic given by 864*a3^2*x0^2 - 2304*a2*a4*x0^2 + 5760*a1*a5*x0^2 + 576*a2*a3*x0*x1 - 3456*a1*a4*x0*x1 + 28800*a0*a5*x0*x1 + 864*a2^2*x1^2 - 2304*a1*a3*x1^2 + 5760*a0*a4*x1^2
>>> from sage.all import * >>> R = QQ['a0, a1, a2, a3, a4, a5, x0, x1']; (a0, a1, a2, a3, a4, a5, x0, x1,) = R._first_ngens(8) >>> f = AlgebraicForm(Integer(2), Integer(5), a0*x1**Integer(5) + a1*x1**Integer(4)*x0 + a2*x1**Integer(3)*x0**Integer(2) ... + a3*x1**Integer(2)*x0**Integer(3) + a4*x1*x0**Integer(4) + a5*x0**Integer(5), x0, x1) >>> transvectant(f, f, Integer(4)) Binary quadratic given by 3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2 + 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1 - 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1 + 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2 + 2/5*a0*a4*x1^2 >>> transvectant(f, f, Integer(4), scale='none') Binary quadratic given by 864*a3^2*x0^2 - 2304*a2*a4*x0^2 + 5760*a1*a5*x0^2 + 576*a2*a3*x0*x1 - 3456*a1*a4*x0*x1 + 28800*a0*a5*x0*x1 + 864*a2^2*x1^2 - 2304*a1*a3*x1^2 + 5760*a0*a4*x1^2
R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[] f = AlgebraicForm(2, 5, a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5, x0, x1) transvectant(f, f, 4) transvectant(f, f, 4, scale='none')If the forms are given as inhomogeneous polynomials, the homogenisation might fail if the polynomial ring has multiple variables. You can circumvent this by making sure the base ring of the polynomial has only one variable:
sage: R.<x,y> = QQ[] sage: quintic = invariant_theory.binary_quintic(x^5 + x^3 + 2*x^2 + y^5, x) sage: transvectant(quintic, quintic, 2) Traceback (most recent call last): ... ValueError: polynomial is not homogeneous sage: R.<y> = QQ[] sage: S.<x> = R[] sage: quintic = invariant_theory.binary_quintic(x^5 + x^3 + 2*x^2 + y^5, x) sage: transvectant(quintic, quintic, 2) Binary sextic given by 1/5*x^6 + 6/5*x^5*h - 3/25*x^4*h^2 + (50*y^5 - 8)/25*x^3*h^3 - 12/25*x^2*h^4 + (3*y^5)/5*x*h^5 + (2*y^5)/5*h^6
>>> from sage.all import * >>> R = QQ['x, y']; (x, y,) = R._first_ngens(2) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + x**Integer(3) + Integer(2)*x**Integer(2) + y**Integer(5), x) >>> transvectant(quintic, quintic, Integer(2)) Traceback (most recent call last): ... ValueError: polynomial is not homogeneous >>> R = QQ['y']; (y,) = R._first_ngens(1) >>> S = R['x']; (x,) = S._first_ngens(1) >>> quintic = invariant_theory.binary_quintic(x**Integer(5) + x**Integer(3) + Integer(2)*x**Integer(2) + y**Integer(5), x) >>> transvectant(quintic, quintic, Integer(2)) Binary sextic given by 1/5*x^6 + 6/5*x^5*h - 3/25*x^4*h^2 + (50*y^5 - 8)/25*x^3*h^3 - 12/25*x^2*h^4 + (3*y^5)/5*x*h^5 + (2*y^5)/5*h^6
R.<x,y> = QQ[] quintic = invariant_theory.binary_quintic(x^5 + x^3 + 2*x^2 + y^5, x) transvectant(quintic, quintic, 2) R.<y> = QQ[] S.<x> = R[] quintic = invariant_theory.binary_quintic(x^5 + x^3 + 2*x^2 + y^5, x) transvectant(quintic, quintic, 2)