Reconstruction of Algebraic Forms

This module reconstructs algebraic forms from the values of their invariants. Given a set of (classical) invariants, it returns a form that attains this values as invariants (up to scaling).

AUTHORS:

• Jesper Noordsij (2018-06): initial version

sage.rings.invariants.reconstruction.binary_cubic_coefficients_from_invariants(discriminant, invariant_choice='default')

Reconstruct a binary cubic from the value of its discriminant.

INPUT:

• discriminant – the value of the discriminant of the binary cubic

• invariant_choice – the type of invariants provided. The accepted options are 'discriminant' and 'default', which are the same. No other options are implemented.

OUTPUT:

A set of coefficients of a binary cubic, whose discriminant is equal to the given discriminant up to a scaling.

EXAMPLES:

Sage

sage: from sage.rings.invariants.reconstruction import binary_cubic_coefficients_from_invariants
sage: coeffs = binary_cubic_coefficients_from_invariants(1)
sage: coeffs
(0, 1, -1, 0)
sage: R.<x> = QQ[]
sage: R(coeffs).discriminant()                                                  # needs sage.libs.pari
1

Python

>>> from sage.all import *
>>> from sage.rings.invariants.reconstruction import binary_cubic_coefficients_from_invariants
>>> coeffs = binary_cubic_coefficients_from_invariants(Integer(1))
>>> coeffs
(0, 1, -1, 0)
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> R(coeffs).discriminant()                                                  # needs sage.libs.pari
1

Sage Live

from sage.rings.invariants.reconstruction import binary_cubic_coefficients_from_invariants
coeffs = binary_cubic_coefficients_from_invariants(1)
coeffs
R.<x> = QQ[]
R(coeffs).discriminant()                                                  # needs sage.libs.pari

The two non-equivalent cubics \(x^3\) and \(x^2*z\) with discriminant 0 can’t be distinguished based on their discriminant, hence an error is raised:

Sage

sage: binary_cubic_coefficients_from_invariants(0)
Traceback (most recent call last):
...
ValueError: no unique reconstruction possible for binary cubics with a double root

Python

>>> from sage.all import *
>>> binary_cubic_coefficients_from_invariants(Integer(0))
Traceback (most recent call last):
...
ValueError: no unique reconstruction possible for binary cubics with a double root

Sage Live

binary_cubic_coefficients_from_invariants(0)

sage.rings.invariants.reconstruction.binary_quadratic_coefficients_from_invariants(discriminant, invariant_choice='default')

Reconstruct a binary quadratic from the value of its discriminant.

INPUT:

• discriminant – the value of the discriminant of the binary quadratic

• invariant_choice – the type of invariants provided. The accepted options are 'discriminant' and 'default', which are the same. No other options are implemented.

OUTPUT:

A set of coefficients of a binary quadratic, whose discriminant is equal to the given discriminant up to a scaling.

EXAMPLES:

Sage

sage: from sage.rings.invariants.reconstruction import binary_quadratic_coefficients_from_invariants
sage: quadratic = invariant_theory.binary_form_from_invariants(2, [24])  # indirect doctest
sage: quadratic
Binary quadratic with coefficients (1, -6, 0)
sage: quadratic.discriminant()
24
sage: binary_quadratic_coefficients_from_invariants(0)
(1, 0, 0)

Python

>>> from sage.all import *
>>> from sage.rings.invariants.reconstruction import binary_quadratic_coefficients_from_invariants
>>> quadratic = invariant_theory.binary_form_from_invariants(Integer(2), [Integer(24)])  # indirect doctest
>>> quadratic
Binary quadratic with coefficients (1, -6, 0)
>>> quadratic.discriminant()
24
>>> binary_quadratic_coefficients_from_invariants(Integer(0))
(1, 0, 0)

Sage Live

from sage.rings.invariants.reconstruction import binary_quadratic_coefficients_from_invariants
quadratic = invariant_theory.binary_form_from_invariants(2, [24])  # indirect doctest
quadratic
quadratic.discriminant()
binary_quadratic_coefficients_from_invariants(0)

sage.rings.invariants.reconstruction.binary_quintic_coefficients_from_invariants(invariants, K=None, invariant_choice='default', scaling='none')

Reconstruct a binary quintic from the values of its (Clebsch) invariants.

INPUT:

• invariants – list or tuple of values of the three or four invariants. The default option requires the Clebsch invariants \(A\), \(B\), \(C\) and \(R\) of the binary quintic.

• K – the field over which the quintic is defined

• invariant_choice – the type of invariants provided. The accepted options are 'clebsch' and 'default', which are the same. No other options are implemented.

• scaling – how the coefficients should be scaled. The accepted values are 'none' for no scaling, 'normalized' to scale in such a way that the resulting coefficients are independent of the scaling of the input invariants and 'coprime' which scales the input invariants by dividing them by their gcd.

OUTPUT:

A set of coefficients of a binary quintic, whose invariants are equal to the given invariants up to a scaling.

EXAMPLES:

First we check the general case, where the invariant \(M\) is nonzero:

Sage

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: invs = quintic.clebsch_invariants(as_tuple=True)
sage: reconstructed = invariant_theory.binary_form_from_invariants(  # indirect doctest
....:     5, invs, variables=quintic.variables())
sage: reconstructed
Binary quintic with coefficients (9592267437341790539005557/244140625000000,
2149296928207625556323004064707/610351562500000000,
11149651890347700974453304786783/76293945312500000,
122650775751894638395648891202734239/47683715820312500000,
323996630945706528474286334593218447/11920928955078125000,
1504506503644608395841632538558481466127/14901161193847656250000)

Python

>>> 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)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> reconstructed = invariant_theory.binary_form_from_invariants(  # indirect doctest
...     Integer(5), invs, variables=quintic.variables())
>>> reconstructed
Binary quintic with coefficients (9592267437341790539005557/244140625000000,
2149296928207625556323004064707/610351562500000000,
11149651890347700974453304786783/76293945312500000,
122650775751894638395648891202734239/47683715820312500000,
323996630945706528474286334593218447/11920928955078125000,
1504506503644608395841632538558481466127/14901161193847656250000)

Sage Live

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)
invs = quintic.clebsch_invariants(as_tuple=True)
reconstructed = invariant_theory.binary_form_from_invariants(  # indirect doctest
    5, invs, variables=quintic.variables())
reconstructed

We can see that the invariants of the reconstructed form match the ones of the original form by scaling the invariants \(B\) and \(C\):

Sage

sage: scale = invs[0]/reconstructed.A_invariant()
sage: invs[1] == reconstructed.B_invariant()*scale^2
True
sage: invs[2] == reconstructed.C_invariant()*scale^3
True

Python

>>> from sage.all import *
>>> scale = invs[Integer(0)]/reconstructed.A_invariant()
>>> invs[Integer(1)] == reconstructed.B_invariant()*scale**Integer(2)
True
>>> invs[Integer(2)] == reconstructed.C_invariant()*scale**Integer(3)
True

Sage Live

scale = invs[0]/reconstructed.A_invariant()
invs[1] == reconstructed.B_invariant()*scale^2
invs[2] == reconstructed.C_invariant()*scale^3

If we compare the form obtained by this reconstruction to the one found by letting the covariants \(\alpha\) and \(\beta\) be the coordinates of the form, we find the forms are the same up to a power of the determinant of \(\alpha\) and \(\beta\):

Sage

sage: alpha = quintic.alpha_covariant()
sage: beta = quintic.beta_covariant()
sage: g = matrix([[alpha(x0=1,x1=0), alpha(x0=0,x1=1)],
....:             [beta(x0=1,x1=0), beta(x0=0,x1=1)]])^-1
sage: transformed = tuple([g.determinant()^-5*x
....:                      for x in quintic.transformed(g).coeffs()])
sage: transformed == reconstructed.coeffs()
True

Python

>>> from sage.all import *
>>> alpha = quintic.alpha_covariant()
>>> beta = quintic.beta_covariant()
>>> g = matrix([[alpha(x0=Integer(1),x1=Integer(0)), alpha(x0=Integer(0),x1=Integer(1))],
...             [beta(x0=Integer(1),x1=Integer(0)), beta(x0=Integer(0),x1=Integer(1))]])**-Integer(1)
>>> transformed = tuple([g.determinant()**-Integer(5)*x
...                      for x in quintic.transformed(g).coeffs()])
>>> transformed == reconstructed.coeffs()
True

Sage Live

alpha = quintic.alpha_covariant()
beta = quintic.beta_covariant()
g = matrix([[alpha(x0=1,x1=0), alpha(x0=0,x1=1)],
            [beta(x0=1,x1=0), beta(x0=0,x1=1)]])^-1
transformed = tuple([g.determinant()^-5*x
                     for x in quintic.transformed(g).coeffs()])
transformed == reconstructed.coeffs()

This can also be seen by computing the \(\alpha\) covariant of the obtained form:

Sage

sage: reconstructed.alpha_covariant().coefficient(x1)
0
sage: reconstructed.alpha_covariant().coefficient(x0) != 0
True

Python

>>> from sage.all import *
>>> reconstructed.alpha_covariant().coefficient(x1)
0
>>> reconstructed.alpha_covariant().coefficient(x0) != Integer(0)
True

Sage Live

reconstructed.alpha_covariant().coefficient(x1)
reconstructed.alpha_covariant().coefficient(x0) != 0

If the invariant \(M\) vanishes, then the coefficients are computed in a different way:

Sage

sage: [A,B,C] = [3,1,2]
sage: M = 2*A*B - 3*C
sage: M
0
sage: from sage.rings.invariants.reconstruction import binary_quintic_coefficients_from_invariants
sage: reconstructed = binary_quintic_coefficients_from_invariants([A,B,C])
sage: reconstructed
(-66741943359375/2097152,
 -125141143798828125/134217728,
 0,
 52793920040130615234375/34359738368,
 19797720015048980712890625/1099511627776,
 -4454487003386020660400390625/17592186044416)
sage: newform = sum([ reconstructed[i]*x0^i*x1^(5-i) for i in range(6) ])
sage: newquintic = invariant_theory.binary_quintic(newform, x0, x1)
sage: scale = 3/newquintic.A_invariant()
sage: [3, newquintic.B_invariant()*scale^2, newquintic.C_invariant()*scale^3]
[3, 1, 2]

Python

>>> from sage.all import *
>>> [A,B,C] = [Integer(3),Integer(1),Integer(2)]
>>> M = Integer(2)*A*B - Integer(3)*C
>>> M
0
>>> from sage.rings.invariants.reconstruction import binary_quintic_coefficients_from_invariants
>>> reconstructed = binary_quintic_coefficients_from_invariants([A,B,C])
>>> reconstructed
(-66741943359375/2097152,
 -125141143798828125/134217728,
 0,
 52793920040130615234375/34359738368,
 19797720015048980712890625/1099511627776,
 -4454487003386020660400390625/17592186044416)
>>> newform = sum([ reconstructed[i]*x0**i*x1**(Integer(5)-i) for i in range(Integer(6)) ])
>>> newquintic = invariant_theory.binary_quintic(newform, x0, x1)
>>> scale = Integer(3)/newquintic.A_invariant()
>>> [Integer(3), newquintic.B_invariant()*scale**Integer(2), newquintic.C_invariant()*scale**Integer(3)]
[3, 1, 2]

Sage Live

[A,B,C] = [3,1,2]
M = 2*A*B - 3*C
M
from sage.rings.invariants.reconstruction import binary_quintic_coefficients_from_invariants
reconstructed = binary_quintic_coefficients_from_invariants([A,B,C])
reconstructed
newform = sum([ reconstructed[i]*x0^i*x1^(5-i) for i in range(6) ])
newquintic = invariant_theory.binary_quintic(newform, x0, x1)
scale = 3/newquintic.A_invariant()
[3, newquintic.B_invariant()*scale^2, newquintic.C_invariant()*scale^3]

Several special cases:

Sage

sage: quintic = invariant_theory.binary_quintic(x0^5 - x1^5, x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 0, 1)
sage: quintic = invariant_theory.binary_quintic(x0*x1*(x0^3-x1^3), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(0, 1, 0, 0, 1, 0)
sage: quintic = invariant_theory.binary_quintic(x0^5 + 10*x0^3*x1^2 - 15*x0*x1^4, x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 10, 0, -15, 0)
sage: quintic = invariant_theory.binary_quintic(x0^2*(x0^3 + x1^3), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 1, 0, 0)
sage: quintic = invariant_theory.binary_quintic(x0*(x0^4 + x1^4), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 1, 0)

Python

>>> from sage.all import *
>>> quintic = invariant_theory.binary_quintic(x0**Integer(5) - x1**Integer(5), x0, x1)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 0, 1)
>>> quintic = invariant_theory.binary_quintic(x0*x1*(x0**Integer(3)-x1**Integer(3)), x0, x1)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> binary_quintic_coefficients_from_invariants(invs)
(0, 1, 0, 0, 1, 0)
>>> quintic = invariant_theory.binary_quintic(x0**Integer(5) + Integer(10)*x0**Integer(3)*x1**Integer(2) - Integer(15)*x0*x1**Integer(4), x0, x1)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> binary_quintic_coefficients_from_invariants(invs)
(1, 0, 10, 0, -15, 0)
>>> quintic = invariant_theory.binary_quintic(x0**Integer(2)*(x0**Integer(3) + x1**Integer(3)), x0, x1)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 1, 0, 0)
>>> quintic = invariant_theory.binary_quintic(x0*(x0**Integer(4) + x1**Integer(4)), x0, x1)
>>> invs = quintic.clebsch_invariants(as_tuple=True)
>>> binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 1, 0)

Sage Live

quintic = invariant_theory.binary_quintic(x0^5 - x1^5, x0, x1)
invs = quintic.clebsch_invariants(as_tuple=True)
binary_quintic_coefficients_from_invariants(invs)
quintic = invariant_theory.binary_quintic(x0*x1*(x0^3-x1^3), x0, x1)
invs = quintic.clebsch_invariants(as_tuple=True)
binary_quintic_coefficients_from_invariants(invs)
quintic = invariant_theory.binary_quintic(x0^5 + 10*x0^3*x1^2 - 15*x0*x1^4, x0, x1)
invs = quintic.clebsch_invariants(as_tuple=True)
binary_quintic_coefficients_from_invariants(invs)
quintic = invariant_theory.binary_quintic(x0^2*(x0^3 + x1^3), x0, x1)
invs = quintic.clebsch_invariants(as_tuple=True)
binary_quintic_coefficients_from_invariants(invs)
quintic = invariant_theory.binary_quintic(x0*(x0^4 + x1^4), x0, x1)
invs = quintic.clebsch_invariants(as_tuple=True)
binary_quintic_coefficients_from_invariants(invs)

For fields of characteristic 2, 3 or 5, there is no reconstruction implemented. This is part of Issue #26786.:

Sage

sage: binary_quintic_coefficients_from_invariants([3,1,2], K=GF(5))
Traceback (most recent call last):
...
NotImplementedError: no reconstruction of binary quintics implemented
for fields of characteristic 2, 3 or 5

Python

>>> from sage.all import *
>>> binary_quintic_coefficients_from_invariants([Integer(3),Integer(1),Integer(2)], K=GF(Integer(5)))
Traceback (most recent call last):
...
NotImplementedError: no reconstruction of binary quintics implemented
for fields of characteristic 2, 3 or 5

Sage Live

binary_quintic_coefficients_from_invariants([3,1,2], K=GF(5))

Make live