Elements of multivariate Laurent polynomial rings¶

class sage.rings.polynomial.laurent_polynomial_mpair.LaurentPolynomial_mpair[source]¶

Bases: LaurentPolynomial

Multivariate Laurent polynomials.

coefficient(mon)[source]¶

Return the coefficient of mon in self, where mon must have the same parent as self.

The coefficient is defined as follows. If \(f\) is this polynomial, then the coefficient \(c_m\) is sum:

\[c_m := \sum_T \frac{T}{m}\]

where the sum is over terms \(T\) in \(f\) that are exactly divisible by \(m\).

A monomial \(m(x,y)\) ‘exactly divides’ \(f(x,y)\) if \(m(x,y) | f(x,y)\) and neither \(x \cdot m(x,y)\) nor \(y \cdot m(x,y)\) divides \(f(x,y)\).

INPUT:

mon – a monomial

OUTPUT: element of the parent of self

Note

To get the constant coefficient, call constant_coefficient().

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)

The coefficient returned is an element of the parent of self; in this case, P.

sage: f = 2 * x * y
sage: c = f.coefficient(x*y); c
2
sage: c.parent()
Multivariate Laurent Polynomial Ring in x, y over Rational Field

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^2 + 5*x*y)*x^-3; f
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2
sage: f.coefficient(y)
5*x^-2
sage: f.coefficient(y^2)
-7*x^-2 + x^-3
sage: f.coefficient(x*y)
0
sage: f.coefficient(x^-2)
-7*y^2 + 5*y
sage: f.coefficient(x^-2*y^2)
-7
sage: f.coefficient(1)
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2

coefficients()[source]¶

Return the nonzero coefficients of self in a list.

The returned list is decreasingly ordered by the term ordering of self.parent().

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ, order='degrevlex')
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.coefficients()
[4, 3, 2, 1]
sage: L.<x,y,z> = LaurentPolynomialRing(QQ,order='lex')
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.coefficients()
[4, 1, 2, 3]

constant_coefficient()[source]¶

Return the constant coefficient of self.

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^2 + 5*x*y)*x^-3; f
-x^6 - 7*x^-2*y^2 + 5*x^-2*y + x^-3*y^2
sage: f.constant_coefficient()
0
sage: f = (x^3 + 2*x^-2*y+y^3)*y^-3; f
x^3*y^-3 + 1 + 2*x^-2*y^-2
sage: f.constant_coefficient()
1

degree(x=None)[source]¶

Return the degree of self.

INPUT:

x – (default: None) a generator of the parent ring

OUTPUT:

If x is None, return the total degree of self. If x is a given generator of the parent ring, the output is the maximum degree of x in self.

EXAMPLES:

sage: R.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.degree()
6
sage: f.degree(x)
7
sage: f.degree(y)
1
sage: f.degree(z)
0

The zero polynomial is defined to have degree \(-\infty\):

sage: R.<x, y, z> = LaurentPolynomialRing(ZZ)
sage: R.zero().degree()
-Infinity
sage: R.zero().degree(x)
-Infinity
sage: R.zero().degree(x) == R.zero().degree(y) == R.zero().degree(z)
True

derivative(*args)[source]¶

The formal derivative of this Laurent polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

sage: R = LaurentPolynomialRing(ZZ,'x, y')
sage: x, y = R.gens()
sage: t = x**4*y + x*y + y + x**(-1) + y**(-3)
sage: t.derivative(x, x)
12*x^2*y + 2*x^-3
sage: t.derivative(y, 2)
12*y^-5

dict()[source]¶: alias of monomial_coefficients().

diff(*args)[source]¶: alias of derivative().

differentiate(*args)[source]¶: alias of derivative().

divides(other)[source]¶

Check if self divides other.

EXAMPLES:

sage: R.<x,y> = LaurentPolynomialRing(QQ)
sage: f1 = x^-2*y^3 - 9 - 1/14*x^-1*y - 1/3*x^-1
sage: h = 3*x^-1 - 3*x^-2*y - 1/2*x^-3*y^2 - x^-3*y + x^-3
sage: f2 = f1 * h
sage: f3 = f2 + x * y
sage: f1.divides(f2)
True
sage: f1.divides(f3)
False
sage: f1.divides(3)
False

Zero is divisible by everything, and only zero is divisible by zero:

sage: R.<x,y> = LaurentPolynomialRing(ZZ)
sage: x.divides(R(0))
True
sage: R(0).divides(x)
False
sage: R(0).divides(R(0))
True

Monomials divide when the exponents allow:

sage: (x*y^-1).divides(x^2*y^-2)
True
sage: (x^2).divides(x)
True

exponents()[source]¶

Return a list of the exponents of self.

EXAMPLES:

sage: L.<w,z> = LaurentPolynomialRing(QQ)
sage: a = w^2*z^-1 + 3; a
w^2*z^-1 + 3
sage: e = a.exponents()
sage: e.sort(); e
[(0, 0), (2, -1)]

factor()[source]¶

Return a Laurent monomial (the unit part of the factorization) and a factored multi-polynomial.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.factor()
(x^3*y^-7*z^-2) * (4*x^4*y^7*z + 3*y^8*z^2 + 2*x*y^7 + x^3*z^2)

has_any_inverse()[source]¶

Return True if self contains any monomials with a negative exponent, False otherwise.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.has_any_inverse()
True
sage: g = x^2 + y^2
sage: g.has_any_inverse()
False

has_inverse_of(i)[source]¶

INPUT:

i – the index of a generator of self.parent()

OUTPUT:

Return True if self contains a monomial including the inverse of self.parent().gen(i), False otherwise.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.has_inverse_of(0)
False
sage: f.has_inverse_of(1)
True
sage: f.has_inverse_of(2)
True

is_constant()[source]¶

Return whether this Laurent polynomial is constant.

EXAMPLES:

sage: L.<a, b> = LaurentPolynomialRing(QQ)
sage: L(0).is_constant()
True
sage: L(42).is_constant()
True
sage: a.is_constant()
False
sage: (1/b).is_constant()
False

is_monomial()[source]¶

Return True if self is a monomial.

EXAMPLES:

sage: k.<y,z> = LaurentPolynomialRing(QQ)
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^-2909).is_monomial()
True
sage: (38*z^-2909).is_monomial()
False

is_square(root=False)[source]¶

Test whether this Laurent polynomial is a square.

INPUT:

root – boolean (default: False); if set to True then return a pair (True, sqrt) with sqrt a square root of this Laurent polynomial when it exists or (False, None)

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: p = 1 + x*y + z^-3
sage: (p**2).is_square()
True
sage: (p**2).is_square(root=True)
(True, x*y + 1 + z^-3)

sage: x.is_square()
False
sage: x.is_square(root=True)
(False, None)

sage: (x**-4 * (1 + z)).is_square(root=False)
False
sage: (x**-4 * (1 + z)).is_square(root=True)
(False, None)

is_unit()[source]¶

Return True if self is a unit.

The ground ring is assumed to be an integral domain.

This means that the Laurent polynomial is a monomial with unit coefficient.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ)
sage: (x*y/2).is_unit()
True
sage: (x + y).is_unit()
False
sage: (L.zero()).is_unit()
False
sage: (L.one()).is_unit()
True

sage: L.<x,y> = LaurentPolynomialRing(ZZ)
sage: (2*x*y).is_unit()
False

is_univariate()[source]¶

Return True if this is a univariate or constant Laurent polynomial, and False otherwise.

EXAMPLES:

sage: R.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = (x^3 + y^-3)*z
sage: f.is_univariate()
False
sage: g = f(1, y, 4)
sage: g.is_univariate()
True
sage: R(1).is_univariate()
True

iterator_exp_coeff()[source]¶

Iterate over self as pairs of (ETuple, coefficient).

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: list(f.iterator_exp_coeff())
[((6, 0), -1), ((-2, 3), -7), ((-2, 1), 5), ((-3, 2), 1)]

monomial_coefficient(mon)[source]¶

Return the coefficient in the base ring of the monomial mon in self, where mon must have the same parent as self.

This function contrasts with the function coefficient() which returns the coefficient of a monomial viewing this polynomial in a polynomial ring over a base ring having fewer variables.

INPUT:

mon – a monomial

See also

For coefficients in a base ring of fewer variables, see coefficient().

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: f.monomial_coefficient(x^-2*y^3)
-7
sage: f.monomial_coefficient(x^2)
0

monomial_coefficients()[source]¶

Return self represented as a dict.

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: sorted(f.monomial_coefficients().items())
[((3, 1, 0), 3), ((4, 0, -2), 2), ((6, -7, 0), 1), ((7, 0, -1), 4)]

dict is an alias:

sage: sorted(f.dict().items())
[((3, 1, 0), 3), ((4, 0, -2), 2), ((6, -7, 0), 1), ((7, 0, -1), 4)]

monomial_reduction()[source]¶

Factor self into a polynomial and a monomial.

OUTPUT:

A tuple (p, v) where p is the underlying polynomial and v is a monomial.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(QQ)
sage: f = y / x + x^2 / y + 3 * x^4 * y^-2
sage: f.monomial_reduction()
(3*x^5 + x^3*y + y^3, x^-1*y^-2)
sage: f = y * x + x^2 / y + 3 * x^4 * y^-2
sage: f.monomial_reduction()
 (3*x^3 + y^3 + x*y, x*y^-2)
sage: x.monomial_reduction()
(1, x)
sage: (y^-1).monomial_reduction()
(1, y^-1)

monomials()[source]¶

Return the list of monomials in self.

EXAMPLES:

sage: P.<x,y> = LaurentPolynomialRing(QQ)
sage: f = (y^2 - x^9 - 7*x*y^3 + 5*x*y)*x^-3
sage: sorted(f.monomials())
[x^-3*y^2, x^-2*y, x^-2*y^3, x^6]

newton_polytope()[source]¶

Return the Newton polytope of this Laurent polynomial.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(QQ)
sage: f = 1 + x*y + y**2 + 33 * x^-3
sage: P = f.newton_polytope(); P                                            # needs sage.geometry.polyhedron
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices

number_of_terms()[source]¶

Return the number of nonzero coefficients of self.

Also called weight, hamming weight or sparsity.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(ZZ)
sage: f = x^3 - y
sage: f.number_of_terms()
2
sage: R(0).number_of_terms()
0
sage: f = (x+1/y)^100
sage: f.number_of_terms()
101

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

quo_rem(right)[source]¶

Divide this Laurent polynomial by right and return a quotient and a remainder.

INPUT:

right – a Laurent polynomial

OUTPUT: a pair of Laurent polynomials

EXAMPLES:

sage: R.<s, t> = LaurentPolynomialRing(QQ)
sage: (s^2 - t^2).quo_rem(s - t)                                            # needs sage.libs.singular
(s + t, 0)
sage: (s^-2 - t^2).quo_rem(s - t)                                           # needs sage.libs.singular
(s + t, -s^2 + s^-2)
sage: (s^-2 - t^2).quo_rem(s^-1 - t)                                        # needs sage.libs.singular
(t + s^-1, 0)

rescale_vars(d, h=None, new_ring=None)[source]¶

Rescale variables in a Laurent polynomial.

INPUT:

d – a dict whose keys are the generator indices and values are the coefficients; so a pair (i, v) means \(x_i \mapsto v x_i\)
h – (optional) a map to be applied to coefficients done after rescaling
new_ring – (optional) a new ring to map the result into

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = x^-2*y + x*y^-2
sage: p.rescale_vars({0: 2, 1: 3})
2/9*x*y^-2 + 3/4*x^-2*y
sage: F = GF(2)
sage: p.rescale_vars({0: 3, 1: 7}, new_ring=L.change_ring(F))
x*y^-2 + x^-2*y

Test for Issue #30331:

sage: F.<z> = CyclotomicField(3)                                            # needs sage.rings.number_field
sage: p.rescale_vars({0: 2, 1: z}, new_ring=L.change_ring(F))               # needs sage.rings.number_field
2*z*x*y^-2 + 1/4*z*x^-2*y

subs(in_dict=None, **kwds)[source]¶

Substitute some variables in this Laurent polynomial.

Variable/value pairs for the substitution may be given as a dictionary or via keyword-value pairs. If both are present, the latter take precedence.

INPUT:

in_dict – dictionary (optional)
**kwds – keyword arguments

OUTPUT: a Laurent polynomial

EXAMPLES:

sage: L.<x, y, z> = LaurentPolynomialRing(QQ)
sage: f = x + 2*y + 3*z
sage: f.subs(x=1)
2*y + 3*z + 1
sage: f.subs(y=1)
x + 3*z + 2
sage: f.subs(z=1)
x + 2*y + 3
sage: f.subs(x=1, y=1, z=1)
6

sage: f = x^-1
sage: f.subs(x=2)
1/2
sage: f.subs({x: 2})
1/2

sage: f = x + 2*y + 3*z
sage: f.subs({x: 1, y: 1, z: 1})
6
sage: f.substitute(x=1, y=1, z=1)
6

toric_coordinate_change(M, h=None, new_ring=None)[source]¶

Apply a matrix to the exponents in a Laurent polynomial.

For efficiency, we implement this directly, rather than as a substitution.

The optional argument h is a map to be applied to coefficients.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = 2*x^2 + y - x*y
sage: p.toric_coordinate_change(Matrix([[1,-3], [1,1]]))
2*x^2*y^2 - x^-2*y^2 + x^-3*y
sage: F = GF(2)
sage: p.toric_coordinate_change(Matrix([[1,-3], [1,1]]),
....:                           new_ring=L.change_ring(F))
x^-2*y^2 + x^-3*y

toric_substitute(v, v1, a, h=None, new_ring=None)[source]¶

Perform a single-variable substitution up to a toric coordinate change.

The optional argument h is a map to be applied to coefficients.

EXAMPLES:

sage: L.<x,y> = LaurentPolynomialRing(QQ, 2)
sage: p = x + y
sage: p.toric_substitute((2,3), (-1,1), 2)
1/2*x^3*y^3 + 2*x^-2*y^-2
sage: F = GF(5)
sage: p.toric_substitute((2,3), (-1,1), 2, new_ring=L.change_ring(F))
3*x^3*y^3 + 2*x^-2*y^-2

univariate_polynomial(R=None)[source]¶

Return a univariate polynomial associated to this multivariate polynomial.

INPUT:

R – (default: None) a univariate Laurent polynomial ring

If this polynomial is not in at most one variable, then a ValueError exception is raised. The new polynomial is over the same base ring as the given LaurentPolynomial and in the variable x if no ring R is provided.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(ZZ)
sage: f = 3*x^2 - 2*y^-1 + 7*x^2*y^2 + 5
sage: f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
sage: g = f(10, y); g
700*y^2 + 305 - 2*y^-1
sage: h = g.univariate_polynomial(); h
-2*y^-1 + 305 + 700*y^2
sage: h.parent()
Univariate Laurent Polynomial Ring in y over Integer Ring
sage: g.univariate_polynomial(LaurentPolynomialRing(QQ,'z'))
-2*z^-1 + 305 + 700*z^2

Here’s an example with a constant multivariate polynomial:

sage: g = R(1)
sage: h = g.univariate_polynomial(); h
1
sage: h.parent()
Univariate Laurent Polynomial Ring in x over Integer Ring

valuation(x=None)[source]¶

Return the valuation of self.

If x is None, the returned valuation is the minimal total degree of the monomials occurring in self. Geometrically, this is the order of vanishing of self at the generic point of the blow-up of the point \((0,0,\ldots,0)\).

If x is not None, then it must be a generator. In that case, the minimum degree of that generator occurring in self is returned. Geometrically, this is the order of vanishing of self at the generic point of the curve \(x = 0\).

INPUT:

x – (optional) a generator; if given, return the valuation with respect to this generator

EXAMPLES:

sage: R.<x,y> = LaurentPolynomialRing(ZZ)
sage: f = 2*x^2*y^-3 - 13*x^-1*y^-3 + 2*x^2*y^-5 - 2*x^-3*y^2
sage: f.valuation()
-4
sage: f.valuation(x)
-3
sage: f.valuation(y)
-5
sage: R.zero().valuation()
+Infinity

variables(sort=True)[source]¶

Return a tuple of all variables occurring in self.

INPUT:

sort – specifies whether the indices shall be sorted

EXAMPLES:

sage: L.<x,y,z> = LaurentPolynomialRing(QQ)
sage: f = 4*x^7*z^-1 + 3*x^3*y + 2*x^4*z^-2 + x^6*y^-7
sage: f.variables()
(z, y, x)
sage: f.variables(sort=False) #random
(y, z, x)