Multivariate Polynomials via libSINGULAR

This module implements specialized and optimized implementations for multivariate polynomials over many coefficient rings, via a shared library interface to SINGULAR. In particular, the following coefficient rings are supported by this implementation:

  • the rational numbers \(\QQ\),

  • the ring of integers \(\ZZ\),

  • \(\ZZ/n\ZZ\) for any integer \(n\),

  • finite fields \(\GF{p^n}\) for \(p\) prime and \(n > 0\),

  • and absolute number fields \(\QQ(a)\).

EXAMPLES:

We show how to construct various multivariate polynomial rings:

sage: P.<x,y,z> = QQ[]
sage: P
Multivariate Polynomial Ring in x, y, z over Rational Field

sage: f = 27/113 * x^2 + y*z + 1/2; f
27/113*x^2 + y*z + 1/2

sage: P.term_order()
Degree reverse lexicographic term order

sage: P = PolynomialRing(GF(127), 3, names='abc', order='lex'); P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
sage: a,b,c = P.gens()
sage: f = 57 * a^2*b + 43 * c + 1; f
57*a^2*b + 43*c + 1
sage: P.term_order()
Lexicographic term order

sage: z = QQ['z'].0
sage: K.<s> = NumberField(z^2 - 2)                                                  # needs sage.rings.number_field
sage: P.<x,y> = PolynomialRing(K, 2)                                                # needs sage.rings.number_field
sage: 1/2*s*x^2 + 3/4*s                                                             # needs sage.rings.number_field
(1/2*s)*x^2 + (3/4*s)

sage: P.<x,y,z> = ZZ[]; P
Multivariate Polynomial Ring in x, y, z over Integer Ring

sage: P.<x,y,z> = Zmod(2^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024

sage: P.<x,y,z> = Zmod(3^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049

sage: P.<x,y,z> = Zmod(2^100)[]; P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 1267650600228229401496703205376

sage: P.<x,y,z> = Zmod(2521352)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>

sage: P.<x,y,z> = Zmod(25213521351515232)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 25213521351515232
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P
Multivariate Polynomial Ring in x, y, z over Rational Field

>>> f = Integer(27)/Integer(113) * x**Integer(2) + y*z + Integer(1)/Integer(2); f
27/113*x^2 + y*z + 1/2

>>> P.term_order()
Degree reverse lexicographic term order

>>> P = PolynomialRing(GF(Integer(127)), Integer(3), names='abc', order='lex'); P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
>>> a,b,c = P.gens()
>>> f = Integer(57) * a**Integer(2)*b + Integer(43) * c + Integer(1); f
57*a^2*b + 43*c + 1
>>> P.term_order()
Lexicographic term order

>>> z = QQ['z'].gen(0)
>>> K = NumberField(z**Integer(2) - Integer(2), names=('s',)); (s,) = K._first_ngens(1)# needs sage.rings.number_field
>>> P = PolynomialRing(K, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field
>>> Integer(1)/Integer(2)*s*x**Integer(2) + Integer(3)/Integer(4)*s                                                             # needs sage.rings.number_field
(1/2*s)*x^2 + (3/4*s)

>>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Integer Ring

>>> P = Zmod(Integer(2)**Integer(10))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024

>>> P = Zmod(Integer(3)**Integer(10))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049

>>> P = Zmod(Integer(2)**Integer(100))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 1267650600228229401496703205376

>>> P = Zmod(Integer(2521352))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
>>> type(P)
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>

>>> P = Zmod(Integer(25213521351515232))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 25213521351515232
>>> type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>

P.<x,y,z> = QQ[]
P
f = 27/113 * x^2 + y*z + 1/2; f
P.term_order()
P = PolynomialRing(GF(127), 3, names='abc', order='lex'); P
a,b,c = P.gens()
f = 57 * a^2*b + 43 * c + 1; f
P.term_order()
z = QQ['z'].0
K.<s> = NumberField(z^2 - 2)                                                  # needs sage.rings.number_field
P.<x,y> = PolynomialRing(K, 2)                                                # needs sage.rings.number_field
1/2*s*x^2 + 3/4*s                                                             # needs sage.rings.number_field
P.<x,y,z> = ZZ[]; P
P.<x,y,z> = Zmod(2^10)[]; P
P.<x,y,z> = Zmod(3^10)[]; P
P.<x,y,z> = Zmod(2^100)[]; P
P.<x,y,z> = Zmod(2521352)[]; P
type(P)
P.<x,y,z> = Zmod(25213521351515232)[]; P
type(P)

We construct the Frobenius morphism on \(\GF{5}[x,y,z]\) over \(\GF{5}\):

sage: R.<x,y,z> = PolynomialRing(GF(5), 3)
sage: frob = R.hom([x^5, y^5, z^5])
sage: frob(x^2 + 2*y - z^4)
-z^20 + x^10 + 2*y^5
sage: frob((x + 2*y)^3)                                                             # needs sage.rings.finite_rings
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
sage: (x^5 + 2*y^5)^3                                                               # needs sage.rings.finite_rings
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(5)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> frob = R.hom([x**Integer(5), y**Integer(5), z**Integer(5)])
>>> frob(x**Integer(2) + Integer(2)*y - z**Integer(4))
-z^20 + x^10 + 2*y^5
>>> frob((x + Integer(2)*y)**Integer(3))                                                             # needs sage.rings.finite_rings
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
>>> (x**Integer(5) + Integer(2)*y**Integer(5))**Integer(3)                                                               # needs sage.rings.finite_rings
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15

R.<x,y,z> = PolynomialRing(GF(5), 3)
frob = R.hom([x^5, y^5, z^5])
frob(x^2 + 2*y - z^4)
frob((x + 2*y)^3)                                                             # needs sage.rings.finite_rings
(x^5 + 2*y^5)^3                                                               # needs sage.rings.finite_rings

We make a polynomial ring in one variable over a polynomial ring in two variables:

sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: S.<t> = PowerSeriesRing(R)
sage: t*(x+y)
(x + y)*t

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> S = PowerSeriesRing(R, names=('t',)); (t,) = S._first_ngens(1)
>>> t*(x+y)
(x + y)*t

R.<x, y> = PolynomialRing(QQ, 2)
S.<t> = PowerSeriesRing(R)
t*(x+y)

Todo

Implement Real, Complex coefficient rings via libSINGULAR

AUTHORS:

  • Martin Albrecht (2007-01): initial implementation

  • Joel Mohler (2008-01): misc improvements, polishing

  • Martin Albrecht (2008-08): added \(\QQ(a)\) and \(\ZZ\) support

  • Simon King (2009-04): improved coercion

  • Martin Albrecht (2009-05): added \(\ZZ/n\ZZ\) support, refactoring

  • Martin Albrecht (2009-06): refactored the code to allow better re-use

  • Simon King (2011-03): use a faster way of conversion from the base ring.

  • Volker Braun (2011-06): major cleanup, refcount singular rings, bugfixes.

class sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular[source]

Bases: MPolynomialRing_base

Construct a multivariate polynomial ring subject to the following conditions:

INPUT:

  • base_ring – base ring (must be either GF(q), ZZ, ZZ/nZZ,

    QQ or absolute number field)

  • n – number of variables (must be at least 1)

  • names – names of ring variables, may be string of list/tuple

  • order – term order (default: degrevlex)

EXAMPLES:

sage: P.<x,y,z> = QQ[]; P
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: f = 27/113 * x^2 + y*z + 1/2; f
27/113*x^2 + y*z + 1/2
sage: P.term_order()
Degree reverse lexicographic term order

sage: P = PolynomialRing(GF(127), 3, names='abc', order='lex'); P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
sage: a,b,c = P.gens()
sage: f = 57 * a^2*b + 43 * c + 1; f
57*a^2*b + 43*c + 1
sage: P.term_order()
Lexicographic term order

sage: z = QQ['z'].0
sage: K.<s> = NumberField(z^2 - 2)                                          # needs sage.rings.number_field
sage: P.<x,y> = PolynomialRing(K, 2)                                        # needs sage.rings.number_field
sage: 1/2*s*x^2 + 3/4*s                                                     # needs sage.rings.number_field
(1/2*s)*x^2 + (3/4*s)

sage: P.<x,y,z> = ZZ[]; P
Multivariate Polynomial Ring in x, y, z over Integer Ring

sage: P.<x,y,z> = Zmod(2^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024

sage: P.<x,y,z> = Zmod(3^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049

sage: P.<x,y,z> = Zmod(2^100)[]; P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 1267650600228229401496703205376

sage: P.<x,y,z> = Zmod(2521352)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>

sage: P.<x,y,z> = Zmod(25213521351515232)[]; P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 25213521351515232
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>

sage: P.<x,y,z> = PolynomialRing(Integers(2^32), order='lex')
sage: P(2^32-1)
4294967295

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Rational Field
>>> f = Integer(27)/Integer(113) * x**Integer(2) + y*z + Integer(1)/Integer(2); f
27/113*x^2 + y*z + 1/2
>>> P.term_order()
Degree reverse lexicographic term order

>>> P = PolynomialRing(GF(Integer(127)), Integer(3), names='abc', order='lex'); P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
>>> a,b,c = P.gens()
>>> f = Integer(57) * a**Integer(2)*b + Integer(43) * c + Integer(1); f
57*a^2*b + 43*c + 1
>>> P.term_order()
Lexicographic term order

>>> z = QQ['z'].gen(0)
>>> K = NumberField(z**Integer(2) - Integer(2), names=('s',)); (s,) = K._first_ngens(1)# needs sage.rings.number_field
>>> P = PolynomialRing(K, Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)# needs sage.rings.number_field
>>> Integer(1)/Integer(2)*s*x**Integer(2) + Integer(3)/Integer(4)*s                                                     # needs sage.rings.number_field
(1/2*s)*x^2 + (3/4*s)

>>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Integer Ring

>>> P = Zmod(Integer(2)**Integer(10))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024

>>> P = Zmod(Integer(3)**Integer(10))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049

>>> P = Zmod(Integer(2)**Integer(100))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 1267650600228229401496703205376

>>> P = Zmod(Integer(2521352))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
>>> type(P)
<class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>

>>> P = Zmod(Integer(25213521351515232))['x, y, z']; (x, y, z,) = P._first_ngens(3); P
Multivariate Polynomial Ring in x, y, z over
 Ring of integers modulo 25213521351515232
>>> type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>

>>> P = PolynomialRing(Integers(Integer(2)**Integer(32)), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> P(Integer(2)**Integer(32)-Integer(1))
4294967295

P.<x,y,z> = QQ[]; P
f = 27/113 * x^2 + y*z + 1/2; f
P.term_order()
P = PolynomialRing(GF(127), 3, names='abc', order='lex'); P
a,b,c = P.gens()
f = 57 * a^2*b + 43 * c + 1; f
P.term_order()
z = QQ['z'].0
K.<s> = NumberField(z^2 - 2)                                          # needs sage.rings.number_field
P.<x,y> = PolynomialRing(K, 2)                                        # needs sage.rings.number_field
1/2*s*x^2 + 3/4*s                                                     # needs sage.rings.number_field
P.<x,y,z> = ZZ[]; P
P.<x,y,z> = Zmod(2^10)[]; P
P.<x,y,z> = Zmod(3^10)[]; P
P.<x,y,z> = Zmod(2^100)[]; P
P.<x,y,z> = Zmod(2521352)[]; P
type(P)
P.<x,y,z> = Zmod(25213521351515232)[]; P
type(P)
P.<x,y,z> = PolynomialRing(Integers(2^32), order='lex')
P(2^32-1)
Element[source]

alias of MPolynomial_libsingular

gen(n=0)[source]

Return the n-th generator of this multivariate polynomial ring.

INPUT:

  • n – nonnegative integer

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.gen(), P.gen(1)
(x, y)

sage: P = PolynomialRing(GF(127), 1000, 'x')
sage: P.gen(500)
x500

sage: P.<SAGE,SINGULAR> = QQ[] # weird names
sage: P.gen(1)
SINGULAR

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.gen(), P.gen(Integer(1))
(x, y)

>>> P = PolynomialRing(GF(Integer(127)), Integer(1000), 'x')
>>> P.gen(Integer(500))
x500

>>> P = QQ['SAGE, SINGULAR']; (SAGE, SINGULAR,) = P._first_ngens(2)# weird names
>>> P.gen(Integer(1))
SINGULAR

P.<x,y,z> = QQ[]
P.gen(), P.gen(1)
P = PolynomialRing(GF(127), 1000, 'x')
P.gen(500)
P.<SAGE,SINGULAR> = QQ[] # weird names
P.gen(1)
ideal(*gens, **kwds)[source]

Create an ideal in this polynomial ring.

INPUT:

  • *gens – list or tuple of generators (or several input arguments)

  • coerce – boolean (default: True); this must be a keyword argument. Only set it to False if you are certain that each generator is already in the ring.

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: sage.rings.ideal.Katsura(P)                                           # needs sage.rings.function_field
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x])
Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> sage.rings.ideal.Katsura(P)                                           # needs sage.rings.function_field
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> P.ideal([x + Integer(2)*y + Integer(2)*z-Integer(1), Integer(2)*x*y + Integer(2)*y*z-y, x**Integer(2) + Integer(2)*y**Integer(2) + Integer(2)*z**Integer(2)-x])
Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x)
 of Multivariate Polynomial Ring in x, y, z over Rational Field

P.<x,y,z> = QQ[]
sage.rings.ideal.Katsura(P)                                           # needs sage.rings.function_field
P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x])
monomial_all_divisors(t)[source]

Return a list of all monomials that divide t.

Coefficients are ignored.

INPUT:

  • t – a monomial

OUTPUT: list of monomials

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.monomial_all_divisors(x^2*z^3)
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.monomial_all_divisors(x**Integer(2)*z**Integer(3))
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]

P.<x,y,z> = QQ[]
P.monomial_all_divisors(x^2*z^3)

ALGORITHM: addwithcarry idea by Toon Segers

monomial_divides(a, b)[source]

Return False if a does not divide b and True otherwise.

Coefficients are ignored.

INPUT:

  • a – monomial

  • b – monomial

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.monomial_divides(x*y*z, x^3*y^2*z^4)
True
sage: P.monomial_divides(x^3*y^2*z^4, x*y*z)
False

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.monomial_divides(x*y*z, x**Integer(3)*y**Integer(2)*z**Integer(4))
True
>>> P.monomial_divides(x**Integer(3)*y**Integer(2)*z**Integer(4), x*y*z)
False

P.<x,y,z> = QQ[]
P.monomial_divides(x*y*z, x^3*y^2*z^4)
P.monomial_divides(x^3*y^2*z^4, x*y*z)
monomial_lcm(f, g)[source]

LCM for monomials. Coefficients are ignored.

INPUT:

  • f – monomial

  • g – monomial

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.monomial_lcm(3/2*x*y,x)
x*y

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.monomial_lcm(Integer(3)/Integer(2)*x*y,x)
x*y

P.<x,y,z> = QQ[]
P.monomial_lcm(3/2*x*y,x)
monomial_pairwise_prime(g, h)[source]

Return True if h and g are pairwise prime. Both are treated as monomials.

Coefficients are ignored.

INPUT:

  • h – monomial

  • g – monomial

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.monomial_pairwise_prime(x^2*z^3, y^4)
True

sage: P.monomial_pairwise_prime(1/2*x^3*y^2, 3/4*y^3)
False

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.monomial_pairwise_prime(x**Integer(2)*z**Integer(3), y**Integer(4))
True

>>> P.monomial_pairwise_prime(Integer(1)/Integer(2)*x**Integer(3)*y**Integer(2), Integer(3)/Integer(4)*y**Integer(3))
False

P.<x,y,z> = QQ[]
P.monomial_pairwise_prime(x^2*z^3, y^4)
P.monomial_pairwise_prime(1/2*x^3*y^2, 3/4*y^3)
monomial_quotient(f, g, coeff=False)[source]

Return f/g, where both f and`` g are treated as monomials.

Coefficients are ignored by default.

INPUT:

  • f – monomial

  • g – monomial

  • coeff – divide coefficients as well (default: False)

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: P.monomial_quotient(3/2*x*y,x)
y

sage: P.monomial_quotient(3/2*x*y,x,coeff=True)
3/2*y

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> P.monomial_quotient(Integer(3)/Integer(2)*x*y,x)
y

>>> P.monomial_quotient(Integer(3)/Integer(2)*x*y,x,coeff=True)
3/2*y

P.<x,y,z> = QQ[]
P.monomial_quotient(3/2*x*y,x)
P.monomial_quotient(3/2*x*y,x,coeff=True)

Note, that \(\ZZ\) behaves different if coeff=True:

sage: P.monomial_quotient(2*x,3*x)
1

sage: P.<x,y> = PolynomialRing(ZZ)
sage: P.monomial_quotient(2*x,3*x,coeff=True)
Traceback (most recent call last):
...
ArithmeticError: Cannot divide these coefficients.

>>> from sage.all import *
>>> P.monomial_quotient(Integer(2)*x,Integer(3)*x)
1

>>> P = PolynomialRing(ZZ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> P.monomial_quotient(Integer(2)*x,Integer(3)*x,coeff=True)
Traceback (most recent call last):
...
ArithmeticError: Cannot divide these coefficients.

P.monomial_quotient(2*x,3*x)
P.<x,y> = PolynomialRing(ZZ)
P.monomial_quotient(2*x,3*x,coeff=True)

Warning

Assumes that the head term of f is a multiple of the head term of g and return the multiplicant m. If this rule is violated, funny things may happen.

monomial_reduce(f, G)[source]

Try to find a g in G where g.lm() divides f. If found (flt,g) is returned, (0,0) otherwise, where flt is f/g.lm().

It is assumed that G is iterable and contains only elements in this polynomial ring.

Coefficients are ignored.

INPUT:

  • f – monomial

  • G – list/set of mpolynomials

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2  ]
sage: P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = x*y**Integer(2)
>>> G = [ Integer(3)/Integer(2)*x**Integer(3) + y**Integer(2) + Integer(1)/Integer(2), Integer(1)/Integer(4)*x*y + Integer(2)/Integer(7), Integer(1)/Integer(2)  ]
>>> P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)

P.<x,y,z> = QQ[]
f = x*y^2
G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2  ]
P.monomial_reduce(f,G)
ngens()[source]

Return the number of variables in this multivariate polynomial ring.

EXAMPLES:

sage: P.<x,y> = QQ[]
sage: P.ngens()
2

sage: k.<a> = GF(2^16)                                                      # needs sage.rings.finite_rings
sage: P = PolynomialRing(k, 1000, 'x')                                      # needs sage.rings.finite_rings
sage: P.ngens()                                                             # needs sage.rings.finite_rings
1000

>>> from sage.all import *
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)
>>> P.ngens()
2

>>> k = GF(Integer(2)**Integer(16), names=('a',)); (a,) = k._first_ngens(1)# needs sage.rings.finite_rings
>>> P = PolynomialRing(k, Integer(1000), 'x')                                      # needs sage.rings.finite_rings
>>> P.ngens()                                                             # needs sage.rings.finite_rings
1000

P.<x,y> = QQ[]
P.ngens()
k.<a> = GF(2^16)                                                      # needs sage.rings.finite_rings
P = PolynomialRing(k, 1000, 'x')                                      # needs sage.rings.finite_rings
P.ngens()                                                             # needs sage.rings.finite_rings
class sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular[source]

Bases: MPolynomial_libsingular

A multivariate polynomial implemented using libSINGULAR.

add_m_mul_q(m, q)[source]

Return self + m*q, where m must be a monomial and q a polynomial.

INPUT:

  • m – a monomial

  • q – a polynomial

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: x.add_m_mul_q(y,z)
y*z + x

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> x.add_m_mul_q(y,z)
y*z + x

P.<x,y,z> = PolynomialRing(QQ,3)
x.add_m_mul_q(y,z)
coefficient(degrees)[source]

Return the coefficient of the variables with the degrees specified in the python dictionary degrees. Mathematically, this is the coefficient in the base ring adjoined by the variables of this ring not listed in degrees. However, the result has the same parent as this polynomial.

This function contrasts with the function monomial_coefficient which returns the coefficient in the base ring of a monomial.

INPUT:

  • degrees – can be any of: - a dictionary of degree restrictions - a list of degree restrictions (with None in the unrestricted variables) - a monomial (very fast, but not as flexible)

OUTPUT: element of the parent of this element

Note

For coefficients of specific monomials, look at monomial_coefficient().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f=x*y+y+5
sage: f.coefficient({x:0,y:1})
1
sage: f.coefficient({x:0})
y + 5
sage: f=(1+y+y^2)*(1+x+x^2)
sage: f.coefficient({x:0})
y^2 + y + 1
sage: f.coefficient([0,None])
y^2 + y + 1
sage: f.coefficient(x)
y^2 + y + 1

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f=x*y+y+Integer(5)
>>> f.coefficient({x:Integer(0),y:Integer(1)})
1
>>> f.coefficient({x:Integer(0)})
y + 5
>>> f=(Integer(1)+y+y**Integer(2))*(Integer(1)+x+x**Integer(2))
>>> f.coefficient({x:Integer(0)})
y^2 + y + 1
>>> f.coefficient([Integer(0),None])
y^2 + y + 1
>>> f.coefficient(x)
y^2 + y + 1

R.<x,y> = QQ[]
f=x*y+y+5
f.coefficient({x:0,y:1})
f.coefficient({x:0})
f=(1+y+y^2)*(1+x+x^2)
f.coefficient({x:0})
f.coefficient([0,None])
f.coefficient(x)

Note that exponents have all variables specified:

sage: x.coefficient(x.exponents()[0])
1
sage: f.coefficient([1,0])
1
sage: f.coefficient({x:1,y:0})
1

>>> from sage.all import *
>>> x.coefficient(x.exponents()[Integer(0)])
1
>>> f.coefficient([Integer(1),Integer(0)])
1
>>> f.coefficient({x:Integer(1),y:Integer(0)})
1

x.coefficient(x.exponents()[0])
f.coefficient([1,0])
f.coefficient({x:1,y:0})

Be aware that this may not be what you think! The physical appearance of the variable x is deceiving – particularly if the exponent would be a variable.

sage: f.coefficient(x^0) # outputs the full polynomial
x^2*y^2 + x^2*y + x*y^2 + x^2 + x*y + y^2 + x + y + 1
sage: R.<x,y> = GF(389)[]
sage: f = x*y + 5
sage: c = f.coefficient({x:0, y:0}); c
5
sage: parent(c)
Multivariate Polynomial Ring in x, y over Finite Field of size 389

>>> from sage.all import *
>>> f.coefficient(x**Integer(0)) # outputs the full polynomial
x^2*y^2 + x^2*y + x*y^2 + x^2 + x*y + y^2 + x + y + 1
>>> R = GF(Integer(389))['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x*y + Integer(5)
>>> c = f.coefficient({x:Integer(0), y:Integer(0)}); c
5
>>> parent(c)
Multivariate Polynomial Ring in x, y over Finite Field of size 389

f.coefficient(x^0) # outputs the full polynomial
R.<x,y> = GF(389)[]
f = x*y + 5
c = f.coefficient({x:0, y:0}); c
parent(c)

AUTHOR:

  • Joel B. Mohler (2007.10.31)

coefficients()[source]

Return the nonzero coefficients of this polynomial in a list. The returned list is decreasingly ordered by the term ordering of the parent.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ, order='degrevlex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[23, 6, 1]

sage: R.<x,y,z> = PolynomialRing(QQ, order='lex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[6, 23, 1]

>>> from sage.all import *
>>> R = PolynomialRing(QQ, order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f=Integer(23)*x**Integer(6)*y**Integer(7) + x**Integer(3)*y+Integer(6)*x**Integer(7)*z
>>> f.coefficients()
[23, 6, 1]

>>> R = PolynomialRing(QQ, order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f=Integer(23)*x**Integer(6)*y**Integer(7) + x**Integer(3)*y+Integer(6)*x**Integer(7)*z
>>> f.coefficients()
[6, 23, 1]

R.<x,y,z> = PolynomialRing(QQ, order='degrevlex')
f=23*x^6*y^7 + x^3*y+6*x^7*z
f.coefficients()
R.<x,y,z> = PolynomialRing(QQ, order='lex')
f=23*x^6*y^7 + x^3*y+6*x^7*z
f.coefficients()

AUTHOR:

  • Didier Deshommes

constant_coefficient()[source]

Return the constant coefficient of this multivariate polynomial.

EXAMPLES:

sage: P.<x, y> = QQ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.constant_coefficient()
5
sage: f = 3*x^2
sage: f.constant_coefficient()
0

>>> from sage.all import *
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.constant_coefficient()
5
>>> f = Integer(3)*x**Integer(2)
>>> f.constant_coefficient()
0

P.<x, y> = QQ[]
f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
f.constant_coefficient()
f = 3*x^2
f.constant_coefficient()
degree(x=None, std_grading=False)[source]

Return the degree of this polynomial.

INPUT:

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

OUTPUT:

If x is None, return the total degree of self. Note that this result is affected by the weighting given to the generators of the parent ring. Otherwise, if x is (or is coercible to) a generator of the parent ring, the output is the maximum degree of x in self. This is not affected by the weighting of the generators.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: f = y^2 - x^9 - x
sage: f.degree(x)
9
sage: f.degree(y)
2
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(x)
3
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(y)
10

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y**Integer(2) - x**Integer(9) - x
>>> f.degree(x)
9
>>> f.degree(y)
2
>>> (y**Integer(10)*x - Integer(7)*x**Integer(2)*y**Integer(5) + Integer(5)*x**Integer(3)).degree(x)
3
>>> (y**Integer(10)*x - Integer(7)*x**Integer(2)*y**Integer(5) + Integer(5)*x**Integer(3)).degree(y)
10

R.<x, y> = QQ[]
f = y^2 - x^9 - x
f.degree(x)
f.degree(y)
(y^10*x - 7*x^2*y^5 + 5*x^3).degree(x)
(y^10*x - 7*x^2*y^5 + 5*x^3).degree(y)

When the generators have a grading (weighting) then the total degree respects this, but the degree for a given generator is unaffected:

sage: T = TermOrder("wdegrevlex", (2, 3))
sage: R.<x, y> = PolynomialRing(QQ, order=T)
sage: f = x^2 * y + y^4
sage: f.degree()
12
sage: f.degree(x)
2
sage: f.degree(y)
4

>>> from sage.all import *
>>> T = TermOrder("wdegrevlex", (Integer(2), Integer(3)))
>>> R = PolynomialRing(QQ, order=T, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) * y + y**Integer(4)
>>> f.degree()
12
>>> f.degree(x)
2
>>> f.degree(y)
4

T = TermOrder("wdegrevlex", (2, 3))
R.<x, y> = PolynomialRing(QQ, order=T)
f = x^2 * y + y^4
f.degree()
f.degree(x)
f.degree(y)

The term ordering of the parent ring determines the grading of the generators.

sage: T = TermOrder('wdegrevlex', (1,2,3,4))
sage: R = PolynomialRing(QQ, 'x', 12, order=T+T+T)
sage: [x.degree() for x in R.gens()]
[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4]

>>> from sage.all import *
>>> T = TermOrder('wdegrevlex', (Integer(1),Integer(2),Integer(3),Integer(4)))
>>> R = PolynomialRing(QQ, 'x', Integer(12), order=T+T+T)
>>> [x.degree() for x in R.gens()]
[1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4]

T = TermOrder('wdegrevlex', (1,2,3,4))
R = PolynomialRing(QQ, 'x', 12, order=T+T+T)
[x.degree() for x in R.gens()]

A matrix term ordering determines the grading of the generators by the first row of the matrix.

sage: m = matrix(3, [3,2,1,1,1,0,1,0,0])
sage: m
[3 2 1]
[1 1 0]
[1 0 0]
sage: R.<x,y,z> = PolynomialRing(QQ, order=TermOrder(m))
sage: x.degree(), y.degree(), z.degree()
(3, 2, 1)
sage: f = x^3*y + x*z^4
sage: f.degree()
11

>>> from sage.all import *
>>> m = matrix(Integer(3), [Integer(3),Integer(2),Integer(1),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0)])
>>> m
[3 2 1]
[1 1 0]
[1 0 0]
>>> R = PolynomialRing(QQ, order=TermOrder(m), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> x.degree(), y.degree(), z.degree()
(3, 2, 1)
>>> f = x**Integer(3)*y + x*z**Integer(4)
>>> f.degree()
11

m = matrix(3, [3,2,1,1,1,0,1,0,0])
m
R.<x,y,z> = PolynomialRing(QQ, order=TermOrder(m))
x.degree(), y.degree(), z.degree()
f = x^3*y + x*z^4
f.degree()

If the first row contains zero, the grading becomes the standard one.

sage: m = matrix(3, [3,0,1,1,1,0,1,0,0])
sage: m
[3 0 1]
[1 1 0]
[1 0 0]
sage: R.<x,y,z> = PolynomialRing(QQ, order=TermOrder(m))
sage: x.degree(), y.degree(), z.degree()
(1, 1, 1)
sage: f = x^3*y + x*z^4
sage: f.degree()
5

>>> from sage.all import *
>>> m = matrix(Integer(3), [Integer(3),Integer(0),Integer(1),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0)])
>>> m
[3 0 1]
[1 1 0]
[1 0 0]
>>> R = PolynomialRing(QQ, order=TermOrder(m), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> x.degree(), y.degree(), z.degree()
(1, 1, 1)
>>> f = x**Integer(3)*y + x*z**Integer(4)
>>> f.degree()
5

m = matrix(3, [3,0,1,1,1,0,1,0,0])
m
R.<x,y,z> = PolynomialRing(QQ, order=TermOrder(m))
x.degree(), y.degree(), z.degree()
f = x^3*y + x*z^4
f.degree()

To get the degree with the standard grading regardless of the term ordering of the parent ring, use std_grading=True.

sage: f.degree(std_grading=True)
5

>>> from sage.all import *
>>> f.degree(std_grading=True)
5

f.degree(std_grading=True)
degrees()[source]

Return a tuple with the maximal degree of each variable in this polynomial. The list of degrees is ordered by the order of the generators.

EXAMPLES:

sage: R.<y0,y1,y2> = PolynomialRing(QQ,3)
sage: q = 3*y0*y1*y1*y2; q
3*y0*y1^2*y2
sage: q.degrees()
(1, 2, 1)
sage: (q + y0^5).degrees()
(5, 2, 1)

>>> from sage.all import *
>>> R = PolynomialRing(QQ,Integer(3), names=('y0', 'y1', 'y2',)); (y0, y1, y2,) = R._first_ngens(3)
>>> q = Integer(3)*y0*y1*y1*y2; q
3*y0*y1^2*y2
>>> q.degrees()
(1, 2, 1)
>>> (q + y0**Integer(5)).degrees()
(5, 2, 1)

R.<y0,y1,y2> = PolynomialRing(QQ,3)
q = 3*y0*y1*y1*y2; q
q.degrees()
(q + y0^5).degrees()
dict()[source]

Return a dictionary representing self. This dictionary is in the same format as the generic MPolynomial: The dictionary consists of ETuple:coefficient pairs.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: f = 2*x*y^3*z^2 + 1/7*x^2 + 2/3
sage: f.monomial_coefficients()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> f = Integer(2)*x*y**Integer(3)*z**Integer(2) + Integer(1)/Integer(7)*x**Integer(2) + Integer(2)/Integer(3)
>>> f.monomial_coefficients()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

R.<x,y,z> = QQ[]
f = 2*x*y^3*z^2 + 1/7*x^2 + 2/3
f.monomial_coefficients()

dict is an alias:

sage: f.dict()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

>>> from sage.all import *
>>> f.dict()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

f.dict()
divides(other)[source]

Return True if this polynomial divides other.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: p = 3*x*y + 2*y*z + x*z
sage: q = x + y + z + 1
sage: r = p * q
sage: p.divides(r)
True
sage: q.divides(p)
False
sage: r.divides(0)
True
sage: R.zero().divides(r)
False
sage: R.zero().divides(0)
True

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> p = Integer(3)*x*y + Integer(2)*y*z + x*z
>>> q = x + y + z + Integer(1)
>>> r = p * q
>>> p.divides(r)
True
>>> q.divides(p)
False
>>> r.divides(Integer(0))
True
>>> R.zero().divides(r)
False
>>> R.zero().divides(Integer(0))
True

R.<x,y,z> = QQ[]
p = 3*x*y + 2*y*z + x*z
q = x + y + z + 1
r = p * q
p.divides(r)
q.divides(p)
r.divides(0)
R.zero().divides(r)
R.zero().divides(0)
exponents(as_ETuples=True)[source]

Return the exponents of the monomials appearing in this polynomial.

INPUT:

  • as_ETuples – boolean (default: True); if True returns the result as a list of ETuples, otherwise returns a list of tuples

EXAMPLES:

sage: R.<a,b,c> = QQ[]
sage: f = a^3 + b + 2*b^2
sage: f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]
sage: f.exponents(as_ETuples=False)
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]

>>> from sage.all import *
>>> R = QQ['a, b, c']; (a, b, c,) = R._first_ngens(3)
>>> f = a**Integer(3) + b + Integer(2)*b**Integer(2)
>>> f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]
>>> f.exponents(as_ETuples=False)
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]

R.<a,b,c> = QQ[]
f = a^3 + b + 2*b^2
f.exponents()
f.exponents(as_ETuples=False)
factor(proof=None)[source]

Return the factorization of this polynomial.

INPUT:

  • proof – ignored

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
x^5 + 2*x^3*y^2 + x^4 + 2*x^2*y^2 + x^3 + 2*x*y^2
sage: F = f.factor(); F
x * (x^2 + x + 1) * (x^2 + 2*y^2)

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = (x**Integer(3) + Integer(2)*y**Integer(2)*x) * (x**Integer(2) + x + Integer(1)); f
x^5 + 2*x^3*y^2 + x^4 + 2*x^2*y^2 + x^3 + 2*x*y^2
>>> F = f.factor(); F
x * (x^2 + x + 1) * (x^2 + 2*y^2)

R.<x, y> = QQ[]
f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
F = f.factor(); F

Next we factor the same polynomial, but over the finite field of order 3.:

sage: R.<x, y> = GF(3)[]
sage: f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
x^5 - x^3*y^2 + x^4 - x^2*y^2 + x^3 - x*y^2
sage: F = f.factor()
sage: F  # order is somewhat random
(-1) * x * (-x + y) * (x + y) * (x - 1)^2

>>> from sage.all import *
>>> R = GF(Integer(3))['x, y']; (x, y,) = R._first_ngens(2)
>>> f = (x**Integer(3) + Integer(2)*y**Integer(2)*x) * (x**Integer(2) + x + Integer(1)); f
x^5 - x^3*y^2 + x^4 - x^2*y^2 + x^3 - x*y^2
>>> F = f.factor()
>>> F  # order is somewhat random
(-1) * x * (-x + y) * (x + y) * (x - 1)^2

R.<x, y> = GF(3)[]
f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
F = f.factor()
F  # order is somewhat random

Next we factor a polynomial, but over a finite field of order 9.:

sage: # needs sage.rings.finite_rings
sage: K.<a> = GF(3^2)
sage: R.<x, y> = K[]
sage: f = (x^3 + 2*a*y^2*x) * (x^2 + x + 1); f
x^5 + (-a)*x^3*y^2 + x^4 + (-a)*x^2*y^2 + x^3 + (-a)*x*y^2
sage: F = f.factor(); F
((-a)) * x * (x - 1)^2 * ((-a + 1)*x^2 + y^2)
sage: f - F
0

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(3)**Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K['x, y']; (x, y,) = R._first_ngens(2)
>>> f = (x**Integer(3) + Integer(2)*a*y**Integer(2)*x) * (x**Integer(2) + x + Integer(1)); f
x^5 + (-a)*x^3*y^2 + x^4 + (-a)*x^2*y^2 + x^3 + (-a)*x*y^2
>>> F = f.factor(); F
((-a)) * x * (x - 1)^2 * ((-a + 1)*x^2 + y^2)
>>> f - F
0

# needs sage.rings.finite_rings
K.<a> = GF(3^2)
R.<x, y> = K[]
f = (x^3 + 2*a*y^2*x) * (x^2 + x + 1); f
F = f.factor(); F
f - F

Next we factor a polynomial over a number field.:

sage: # needs sage.rings.number_field
sage: p = polygen(ZZ, 'p')
sage: K.<s> = NumberField(p^3 - 2)
sage: KXY.<x,y> = K[]
sage: factor(x^3 - 2*y^3)
(x + (-s)*y) * (x^2 + s*x*y + (s^2)*y^2)
sage: k = (x^3-2*y^3)^5*(x+s*y)^2*(2/3 + s^2)
sage: k.factor()
((s^2 + 2/3)) * (x + s*y)^2 * (x + (-s)*y)^5 * (x^2 + s*x*y + (s^2)*y^2)^5

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> p = polygen(ZZ, 'p')
>>> K = NumberField(p**Integer(3) - Integer(2), names=('s',)); (s,) = K._first_ngens(1)
>>> KXY = K['x, y']; (x, y,) = KXY._first_ngens(2)
>>> factor(x**Integer(3) - Integer(2)*y**Integer(3))
(x + (-s)*y) * (x^2 + s*x*y + (s^2)*y^2)
>>> k = (x**Integer(3)-Integer(2)*y**Integer(3))**Integer(5)*(x+s*y)**Integer(2)*(Integer(2)/Integer(3) + s**Integer(2))
>>> k.factor()
((s^2 + 2/3)) * (x + s*y)^2 * (x + (-s)*y)^5 * (x^2 + s*x*y + (s^2)*y^2)^5

# needs sage.rings.number_field
p = polygen(ZZ, 'p')
K.<s> = NumberField(p^3 - 2)
KXY.<x,y> = K[]
factor(x^3 - 2*y^3)
k = (x^3-2*y^3)^5*(x+s*y)^2*(2/3 + s^2)
k.factor()

This shows that issue Issue #2780 is fixed, i.e. that the unit part of the factorization is set correctly:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 + 1)
sage: R.<y, z> = PolynomialRing(K)
sage: f = 2*y^2 + 2*z^2
sage: F = f.factor(); F.unit()
2

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('y', 'z',)); (y, z,) = R._first_ngens(2)
>>> f = Integer(2)*y**Integer(2) + Integer(2)*z**Integer(2)
>>> F = f.factor(); F.unit()
2

# needs sage.rings.number_field
x = polygen(ZZ, 'x')
K.<a> = NumberField(x^2 + 1)
R.<y, z> = PolynomialRing(K)
f = 2*y^2 + 2*z^2
F = f.factor(); F.unit()

Another example:

sage: R.<x,y,z> = GF(32003)[]                                               # needs sage.rings.finite_rings
sage: f = 9*(x-1)^2*(y+z)                                                   # needs sage.rings.finite_rings
sage: f.factor()                                                            # needs sage.rings.finite_rings
(9) * (y + z) * (x - 1)^2

sage: R.<x,w,v,u> = QQ['x','w','v','u']
sage: p = (4*v^4*u^2 - 16*v^2*u^4 + 16*u^6 - 4*v^4*u + 8*v^2*u^3 + v^4)
sage: p.factor()
(-2*v^2*u + 4*u^3 + v^2)^2
sage: R.<a,b,c,d> = QQ[]
sage: f =  (-2) * (a - d) * (-a + b) * (b - d) * (a - c) * (b - c) * (c - d)
sage: F = f.factor(); F
(-2) * (c - d) * (-b + c) * (b - d) * (-a + c) * (-a + b) * (a - d)
sage: F[0][0]
c - d
sage: F.unit()
-2

>>> from sage.all import *
>>> R = GF(Integer(32003))['x, y, z']; (x, y, z,) = R._first_ngens(3)# needs sage.rings.finite_rings
>>> f = Integer(9)*(x-Integer(1))**Integer(2)*(y+z)                                                   # needs sage.rings.finite_rings
>>> f.factor()                                                            # needs sage.rings.finite_rings
(9) * (y + z) * (x - 1)^2

>>> R = QQ['x','w','v','u']; (x, w, v, u,) = R._first_ngens(4)
>>> p = (Integer(4)*v**Integer(4)*u**Integer(2) - Integer(16)*v**Integer(2)*u**Integer(4) + Integer(16)*u**Integer(6) - Integer(4)*v**Integer(4)*u + Integer(8)*v**Integer(2)*u**Integer(3) + v**Integer(4))
>>> p.factor()
(-2*v^2*u + 4*u^3 + v^2)^2
>>> R = QQ['a, b, c, d']; (a, b, c, d,) = R._first_ngens(4)
>>> f =  (-Integer(2)) * (a - d) * (-a + b) * (b - d) * (a - c) * (b - c) * (c - d)
>>> F = f.factor(); F
(-2) * (c - d) * (-b + c) * (b - d) * (-a + c) * (-a + b) * (a - d)
>>> F[Integer(0)][Integer(0)]
c - d
>>> F.unit()
-2

R.<x,y,z> = GF(32003)[]                                               # needs sage.rings.finite_rings
f = 9*(x-1)^2*(y+z)                                                   # needs sage.rings.finite_rings
f.factor()                                                            # needs sage.rings.finite_rings
R.<x,w,v,u> = QQ['x','w','v','u']
p = (4*v^4*u^2 - 16*v^2*u^4 + 16*u^6 - 4*v^4*u + 8*v^2*u^3 + v^4)
p.factor()
R.<a,b,c,d> = QQ[]
f =  (-2) * (a - d) * (-a + b) * (b - d) * (a - c) * (b - c) * (c - d)
F = f.factor(); F
F[0][0]
F.unit()

Constant elements are factorized in the base rings.

sage: P.<x,y> = ZZ[]
sage: P(2^3*7).factor()
2^3 * 7
sage: P.<x,y> = GF(2)[]
sage: P(1).factor()
1

>>> from sage.all import *
>>> P = ZZ['x, y']; (x, y,) = P._first_ngens(2)
>>> P(Integer(2)**Integer(3)*Integer(7)).factor()
2^3 * 7
>>> P = GF(Integer(2))['x, y']; (x, y,) = P._first_ngens(2)
>>> P(Integer(1)).factor()
1

P.<x,y> = ZZ[]
P(2^3*7).factor()
P.<x,y> = GF(2)[]
P(1).factor()

Factorization for finite prime fields with characteristic \(> 2^{29}\) is not supported

sage: q = 1073741789
sage: T.<aa, bb> = PolynomialRing(GF(q))                                    # needs sage.rings.finite_rings
sage: f = aa^2 + 12124343*bb*aa + 32434598*bb^2                             # needs sage.rings.finite_rings
sage: f.factor()                                                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials
over prime fields with characteristic > 2^29 is not implemented.

>>> from sage.all import *
>>> q = Integer(1073741789)
>>> T = PolynomialRing(GF(q), names=('aa', 'bb',)); (aa, bb,) = T._first_ngens(2)# needs sage.rings.finite_rings
>>> f = aa**Integer(2) + Integer(12124343)*bb*aa + Integer(32434598)*bb**Integer(2)                             # needs sage.rings.finite_rings
>>> f.factor()                                                            # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials
over prime fields with characteristic > 2^29 is not implemented.

q = 1073741789
T.<aa, bb> = PolynomialRing(GF(q))                                    # needs sage.rings.finite_rings
f = aa^2 + 12124343*bb*aa + 32434598*bb^2                             # needs sage.rings.finite_rings
f.factor()                                                            # needs sage.rings.finite_rings

Factorization over the integers is now supported, see Issue #17840:

sage: P.<x,y> = PolynomialRing(ZZ)
sage: f = 12 * (3*x*y + 4) * (5*x - 2) * (2*y + 7)^2
sage: f.factor()
2^2 * 3 * (2*y + 7)^2 * (5*x - 2) * (3*x*y + 4)
sage: g = -12 * (x^2 - y^2)
sage: g.factor()
(-1) * 2^2 * 3 * (x - y) * (x + y)
sage: factor(-4*x*y - 2*x + 2*y + 1)
(-1) * (2*y + 1) * (2*x - 1)

>>> from sage.all import *
>>> P = PolynomialRing(ZZ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = Integer(12) * (Integer(3)*x*y + Integer(4)) * (Integer(5)*x - Integer(2)) * (Integer(2)*y + Integer(7))**Integer(2)
>>> f.factor()
2^2 * 3 * (2*y + 7)^2 * (5*x - 2) * (3*x*y + 4)
>>> g = -Integer(12) * (x**Integer(2) - y**Integer(2))
>>> g.factor()
(-1) * 2^2 * 3 * (x - y) * (x + y)
>>> factor(-Integer(4)*x*y - Integer(2)*x + Integer(2)*y + Integer(1))
(-1) * (2*y + 1) * (2*x - 1)

P.<x,y> = PolynomialRing(ZZ)
f = 12 * (3*x*y + 4) * (5*x - 2) * (2*y + 7)^2
f.factor()
g = -12 * (x^2 - y^2)
g.factor()
factor(-4*x*y - 2*x + 2*y + 1)

Factorization over non-integral domains is not supported

sage: R.<x,y> = PolynomialRing(Zmod(4))
sage: f = (2*x + 1) * (x^2 + x + 1)
sage: f.factor()
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials
over Ring of integers modulo 4 is not implemented.

>>> from sage.all import *
>>> R = PolynomialRing(Zmod(Integer(4)), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = (Integer(2)*x + Integer(1)) * (x**Integer(2) + x + Integer(1))
>>> f.factor()
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials
over Ring of integers modulo 4 is not implemented.

R.<x,y> = PolynomialRing(Zmod(4))
f = (2*x + 1) * (x^2 + x + 1)
f.factor()
gcd(right, algorithm=None, **kwds)[source]

Return the greatest common divisor of self and right.

INPUT:

  • right – polynomial

  • algorithm - 'ezgcd' – EZGCD algorithm - 'modular' – multi-modular algorithm (default)

  • **kwds – ignored

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: f = (x*y*z)^6 - 1
sage: g = (x*y*z)^4 - 1
sage: f.gcd(g)
x^2*y^2*z^2 - 1
sage: GCD([x^3 - 3*x + 2, x^4 - 1, x^6 -1])
x - 1

sage: R.<x,y> = QQ[]
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)
x^2

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = (x*y*z)**Integer(6) - Integer(1)
>>> g = (x*y*z)**Integer(4) - Integer(1)
>>> f.gcd(g)
x^2*y^2*z^2 - 1
>>> GCD([x**Integer(3) - Integer(3)*x + Integer(2), x**Integer(4) - Integer(1), x**Integer(6) -Integer(1)])
x - 1

>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = (x**Integer(3) + Integer(2)*y**Integer(2)*x)**Integer(2)
>>> g = x**Integer(2)*y**Integer(2)
>>> f.gcd(g)
x^2

P.<x,y,z> = QQ[]
f = (x*y*z)^6 - 1
g = (x*y*z)^4 - 1
f.gcd(g)
GCD([x^3 - 3*x + 2, x^4 - 1, x^6 -1])
R.<x,y> = QQ[]
f = (x^3 + 2*y^2*x)^2
g = x^2*y^2
f.gcd(g)

We compute a gcd over a finite field:

sage: # needs sage.rings.finite_rings
sage: F.<u> = GF(31^2)
sage: R.<x,y,z> = F[]
sage: p = x^3 + (1+u)*y^3 + z^3
sage: q = p^3 * (x - y + z*u)
sage: gcd(p,q)
x^3 + (u + 1)*y^3 + z^3
sage: gcd(p,q)  # yes, twice -- tests that singular ring is properly set.
x^3 + (u + 1)*y^3 + z^3

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> F = GF(Integer(31)**Integer(2), names=('u',)); (u,) = F._first_ngens(1)
>>> R = F['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> p = x**Integer(3) + (Integer(1)+u)*y**Integer(3) + z**Integer(3)
>>> q = p**Integer(3) * (x - y + z*u)
>>> gcd(p,q)
x^3 + (u + 1)*y^3 + z^3
>>> gcd(p,q)  # yes, twice -- tests that singular ring is properly set.
x^3 + (u + 1)*y^3 + z^3

# needs sage.rings.finite_rings
F.<u> = GF(31^2)
R.<x,y,z> = F[]
p = x^3 + (1+u)*y^3 + z^3
q = p^3 * (x - y + z*u)
gcd(p,q)
gcd(p,q)  # yes, twice -- tests that singular ring is properly set.

We compute a gcd over a number field:

sage: # needs sage.rings.number_field
sage: x = polygen(QQ)
sage: F.<u> = NumberField(x^3 - 2)
sage: R.<x,y,z> = F[]
sage: p = x^3 + (1+u)*y^3 + z^3
sage: q = p^3 * (x - y + z*u)
sage: gcd(p,q)
x^3 + (u + 1)*y^3 + z^3

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(QQ)
>>> F = NumberField(x**Integer(3) - Integer(2), names=('u',)); (u,) = F._first_ngens(1)
>>> R = F['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> p = x**Integer(3) + (Integer(1)+u)*y**Integer(3) + z**Integer(3)
>>> q = p**Integer(3) * (x - y + z*u)
>>> gcd(p,q)
x^3 + (u + 1)*y^3 + z^3

# needs sage.rings.number_field
x = polygen(QQ)
F.<u> = NumberField(x^3 - 2)
R.<x,y,z> = F[]
p = x^3 + (1+u)*y^3 + z^3
q = p^3 * (x - y + z*u)
gcd(p,q)
global_height(prec=None)[source]

Return the (projective) global height of the polynomial.

This returns the absolute logarithmic height of the coefficients thought of as a projective point.

INPUT:

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 3*x^3 + 2*x*y^2
sage: exp(f.global_height())                                                # needs sage.symbolic
3.00000000000000

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(3) + Integer(2)*x*y**Integer(2)
>>> exp(f.global_height())                                                # needs sage.symbolic
3.00000000000000

R.<x,y> = PolynomialRing(QQ)
f = 3*x^3 + 2*x*y^2
exp(f.global_height())                                                # needs sage.symbolic

sage: # needs sage.rings.number_field
sage: K.<k> = CyclotomicField(3)
sage: R.<x,y> = PolynomialRing(K, sparse=True)
sage: f = k*x*y + 1
sage: exp(f.global_height())
1.00000000000000

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(3), names=('k',)); (k,) = K._first_ngens(1)
>>> R = PolynomialRing(K, sparse=True, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = k*x*y + Integer(1)
>>> exp(f.global_height())
1.00000000000000

# needs sage.rings.number_field
K.<k> = CyclotomicField(3)
R.<x,y> = PolynomialRing(K, sparse=True)
f = k*x*y + 1
exp(f.global_height())

Scaling should not change the result:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 1/25*x^2 + 25/3*x*y + y^2
sage: f.global_height()                                                     # needs sage.symbolic
6.43775164973640
sage: g = 100 * f
sage: g.global_height()                                                     # needs sage.symbolic
6.43775164973640

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(25)*x**Integer(2) + Integer(25)/Integer(3)*x*y + y**Integer(2)
>>> f.global_height()                                                     # needs sage.symbolic
6.43775164973640
>>> g = Integer(100) * f
>>> g.global_height()                                                     # needs sage.symbolic
6.43775164973640

R.<x,y> = PolynomialRing(QQ)
f = 1/25*x^2 + 25/3*x*y + y^2
f.global_height()                                                     # needs sage.symbolic
g = 100 * f
g.global_height()                                                     # needs sage.symbolic

sage: R.<x> = PolynomialRing(QQ)
sage: K.<k> = NumberField(x^2 + 5)
sage: T.<t,w> = PolynomialRing(K)
sage: f = 1/1331 * t^2 + 5 * w + 7
sage: f.global_height()                                                     # needs sage.symbolic
9.13959596745043

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(5), names=('k',)); (k,) = K._first_ngens(1)
>>> T = PolynomialRing(K, names=('t', 'w',)); (t, w,) = T._first_ngens(2)
>>> f = Integer(1)/Integer(1331) * t**Integer(2) + Integer(5) * w + Integer(7)
>>> f.global_height()                                                     # needs sage.symbolic
9.13959596745043

R.<x> = PolynomialRing(QQ)
K.<k> = NumberField(x^2 + 5)
T.<t,w> = PolynomialRing(K)
f = 1/1331 * t^2 + 5 * w + 7
f.global_height()                                                     # needs sage.symbolic

sage: R.<x,y> = QQ[]
sage: f = 1/123*x*y + 12
sage: f.global_height(prec=2)                                               # needs sage.symbolic
8.0

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(123)*x*y + Integer(12)
>>> f.global_height(prec=Integer(2))                                               # needs sage.symbolic
8.0

R.<x,y> = QQ[]
f = 1/123*x*y + 12
f.global_height(prec=2)                                               # needs sage.symbolic

sage: R.<x,y> = QQ[]
sage: f = 0*x*y
sage: f.global_height()
0.000000000000000

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(0)*x*y
>>> f.global_height()
0.000000000000000

R.<x,y> = QQ[]
f = 0*x*y
f.global_height()
gradient()[source]

Return a list of partial derivatives of this polynomial, ordered by the variables of the parent.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: f= x*y + 1
sage: f.gradient()
[y, x, 0]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> f= x*y + Integer(1)
>>> f.gradient()
[y, x, 0]

P.<x,y,z> = PolynomialRing(QQ,3)
f= x*y + 1
f.gradient()
hamming_weight()[source]

Return the number of nonzero coefficients of this polynomial.

This is also called weight, hamming_weight() or sparsity.

EXAMPLES:

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

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(3) - y
>>> f.number_of_terms()
2
>>> R(Integer(0)).number_of_terms()
0
>>> f = (x+y)**Integer(100)
>>> f.number_of_terms()
101

R.<x, y> = ZZ[]
f = x^3 - y
f.number_of_terms()
R(0).number_of_terms()
f = (x+y)^100
f.number_of_terms()

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

>>> from sage.all import *
>>> f.hamming_weight()
101

f.hamming_weight()
in_subalgebra(J, algorithm='algebra_containment')[source]

Return whether this polynomial is contained in the subalgebra generated by J

INPUT:

  • J – list of elements of the parent polynomial ring

  • algorithm – can be 'algebra_containment' (the default), 'inSubring', or 'groebner'

    • 'algebra_containment': use Singular’s algebra_containment function, https://www.singular.uni-kl.de/Manual/4-2-1/sing_1247.htm#SEC1328. The Singular documentation suggests that this is frequently faster than the next option.

    • 'inSubring': use Singular’s inSubring function, https://www.singular.uni-kl.de/Manual/4-2-0/sing_1240.htm#SEC1321.

    • 'groebner': use the algorithm described in Singular’s documentation, but within Sage: if the subalgebra generators are \(y_1\), …, \(y_m\), then create a new polynomial algebra with the old generators along with new ones: \(z_1\), …, \(z_m\). Create the ideal \((z_1 - y_1, ..., z_m - y_m)\), and reduce the polynomial modulo this ideal. The polynomial is contained in the subalgebra if and only if the remainder involves only the new variables \(z_i\).

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: J = [x^2 + y^2, x^2 + z^2]
sage: (y^2).in_subalgebra(J)
False
sage: a = (x^2 + y^2) * (x^2 + z^2)
sage: a.in_subalgebra(J, algorithm='inSubring')
True
sage: (a^2).in_subalgebra(J, algorithm='groebner')
True
sage: (a + a^2).in_subalgebra(J)
True

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> J = [x**Integer(2) + y**Integer(2), x**Integer(2) + z**Integer(2)]
>>> (y**Integer(2)).in_subalgebra(J)
False
>>> a = (x**Integer(2) + y**Integer(2)) * (x**Integer(2) + z**Integer(2))
>>> a.in_subalgebra(J, algorithm='inSubring')
True
>>> (a**Integer(2)).in_subalgebra(J, algorithm='groebner')
True
>>> (a + a**Integer(2)).in_subalgebra(J)
True

P.<x,y,z> = QQ[]
J = [x^2 + y^2, x^2 + z^2]
(y^2).in_subalgebra(J)
a = (x^2 + y^2) * (x^2 + z^2)
a.in_subalgebra(J, algorithm='inSubring')
(a^2).in_subalgebra(J, algorithm='groebner')
(a + a^2).in_subalgebra(J)
integral(var)[source]

Integrate this polynomial with respect to the provided variable.

One requires that \(\QQ\) is contained in the ring.

INPUT:

  • variable – the integral is taken with respect to variable

EXAMPLES:

sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: f = 3*x^3*y^2 + 5*y^2 + 3*x + 2
sage: f.integral(x)
3/4*x^4*y^2 + 5*x*y^2 + 3/2*x^2 + 2*x
sage: f.integral(y)
x^3*y^3 + 5/3*y^3 + 3*x*y + 2*y

>>> from sage.all import *
>>> R = PolynomialRing(QQ, Integer(2), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(3)*y**Integer(2) + Integer(5)*y**Integer(2) + Integer(3)*x + Integer(2)
>>> f.integral(x)
3/4*x^4*y^2 + 5*x*y^2 + 3/2*x^2 + 2*x
>>> f.integral(y)
x^3*y^3 + 5/3*y^3 + 3*x*y + 2*y

R.<x, y> = PolynomialRing(QQ, 2)
f = 3*x^3*y^2 + 5*y^2 + 3*x + 2
f.integral(x)
f.integral(y)

Check that Issue #15896 is solved:

sage: s = x+y
sage: s.integral(x)+x
1/2*x^2 + x*y + x
sage: s.integral(x)*s
1/2*x^3 + 3/2*x^2*y + x*y^2

>>> from sage.all import *
>>> s = x+y
>>> s.integral(x)+x
1/2*x^2 + x*y + x
>>> s.integral(x)*s
1/2*x^3 + 3/2*x^2*y + x*y^2

s = x+y
s.integral(x)+x
s.integral(x)*s
inverse_of_unit()[source]

Return the inverse of this polynomial if it is a unit.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: x.inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: Element is not a unit.

sage: R(1/2).inverse_of_unit()
2

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> x.inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: Element is not a unit.

>>> R(Integer(1)/Integer(2)).inverse_of_unit()
2

R.<x,y> = QQ[]
x.inverse_of_unit()
R(1/2).inverse_of_unit()
is_constant()[source]

Return True if this polynomial is constant.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(127))
sage: x.is_constant()
False
sage: P(1).is_constant()
True

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(127)), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> x.is_constant()
False
>>> P(Integer(1)).is_constant()
True

P.<x,y,z> = PolynomialRing(GF(127))
x.is_constant()
P(1).is_constant()
is_homogeneous()[source]

Return True if this polynomial is homogeneous.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(RationalField(), 2)
sage: (x+y).is_homogeneous()
True
sage: (x.parent()(0)).is_homogeneous()
True
sage: (x+y^2).is_homogeneous()
False
sage: (x^2 + y^2).is_homogeneous()
True
sage: (x^2 + y^2*x).is_homogeneous()
False
sage: (x^2*y + y^2*x).is_homogeneous()
True

>>> from sage.all import *
>>> P = PolynomialRing(RationalField(), Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> (x+y).is_homogeneous()
True
>>> (x.parent()(Integer(0))).is_homogeneous()
True
>>> (x+y**Integer(2)).is_homogeneous()
False
>>> (x**Integer(2) + y**Integer(2)).is_homogeneous()
True
>>> (x**Integer(2) + y**Integer(2)*x).is_homogeneous()
False
>>> (x**Integer(2)*y + y**Integer(2)*x).is_homogeneous()
True

P.<x,y> = PolynomialRing(RationalField(), 2)
(x+y).is_homogeneous()
(x.parent()(0)).is_homogeneous()
(x+y^2).is_homogeneous()
(x^2 + y^2).is_homogeneous()
(x^2 + y^2*x).is_homogeneous()
(x^2*y + y^2*x).is_homogeneous()
is_monomial()[source]

Return True if this polynomial is a monomial. A monomial is defined to be a product of generators with coefficient 1.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: x.is_monomial()
True
sage: (2*x).is_monomial()
False
sage: (x*y).is_monomial()
True
sage: (x*y + x).is_monomial()
False
sage: P(2).is_monomial()
False
sage: P.zero().is_monomial()
False

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> x.is_monomial()
True
>>> (Integer(2)*x).is_monomial()
False
>>> (x*y).is_monomial()
True
>>> (x*y + x).is_monomial()
False
>>> P(Integer(2)).is_monomial()
False
>>> P.zero().is_monomial()
False

P.<x,y,z> = PolynomialRing(QQ)
x.is_monomial()
(2*x).is_monomial()
(x*y).is_monomial()
(x*y + x).is_monomial()
P(2).is_monomial()
P.zero().is_monomial()
is_squarefree()[source]

Return True if this polynomial is square free.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: f= x^2 + 2*x*y + 1/2*z
sage: f.is_squarefree()
True
sage: h = f^2
sage: h.is_squarefree()
False

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> f= x**Integer(2) + Integer(2)*x*y + Integer(1)/Integer(2)*z
>>> f.is_squarefree()
True
>>> h = f**Integer(2)
>>> h.is_squarefree()
False

P.<x,y,z> = PolynomialRing(QQ)
f= x^2 + 2*x*y + 1/2*z
f.is_squarefree()
h = f^2
h.is_squarefree()
is_term()[source]

Return True if self is a term, which we define to be a product of generators times some coefficient, which need not be 1.

Use is_monomial() to require that the coefficient be 1.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ)
sage: x.is_term()
True
sage: (2*x).is_term()
True
sage: (x*y).is_term()
True
sage: (x*y + x).is_term()
False
sage: P(2).is_term()
True
sage: P.zero().is_term()
False

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> x.is_term()
True
>>> (Integer(2)*x).is_term()
True
>>> (x*y).is_term()
True
>>> (x*y + x).is_term()
False
>>> P(Integer(2)).is_term()
True
>>> P.zero().is_term()
False

P.<x,y,z> = PolynomialRing(QQ)
x.is_term()
(2*x).is_term()
(x*y).is_term()
(x*y + x).is_term()
P(2).is_term()
P.zero().is_term()
is_univariate()[source]

Return True if self is a univariate polynomial, that is if self contains only one variable.

EXAMPLES:

sage: P.<x,y,z> = GF(2)[]
sage: f = x^2 + 1
sage: f.is_univariate()
True
sage: f = y*x^2 + 1
sage: f.is_univariate()
False
sage: f = P(0)
sage: f.is_univariate()
True

>>> from sage.all import *
>>> P = GF(Integer(2))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = x**Integer(2) + Integer(1)
>>> f.is_univariate()
True
>>> f = y*x**Integer(2) + Integer(1)
>>> f.is_univariate()
False
>>> f = P(Integer(0))
>>> f.is_univariate()
True

P.<x,y,z> = GF(2)[]
f = x^2 + 1
f.is_univariate()
f = y*x^2 + 1
f.is_univariate()
f = P(0)
f.is_univariate()
is_zero()[source]

Return True if this polynomial is zero.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ)
sage: x.is_zero()
False
sage: (x - x).is_zero()
True

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> x.is_zero()
False
>>> (x - x).is_zero()
True

P.<x,y> = PolynomialRing(QQ)
x.is_zero()
(x - x).is_zero()
iterator_exp_coeff(as_ETuples=True)[source]

Iterate over self as pairs of ((E)Tuple, coefficient).

INPUT:

  • as_ETuples – boolean (default: True); if True iterate over pairs whose first element is an ETuple, otherwise as a tuples

EXAMPLES:

sage: R.<a,b,c> = QQ[]
sage: f = a*c^3 + a^2*b + 2*b^4
sage: list(f.iterator_exp_coeff())
[((0, 4, 0), 2), ((1, 0, 3), 1), ((2, 1, 0), 1)]
sage: list(f.iterator_exp_coeff(as_ETuples=False))
[((0, 4, 0), 2), ((1, 0, 3), 1), ((2, 1, 0), 1)]

sage: R.<a,b,c> = PolynomialRing(QQ, 3, order='lex')
sage: f = a*c^3 + a^2*b + 2*b^4
sage: list(f.iterator_exp_coeff())
[((2, 1, 0), 1), ((1, 0, 3), 1), ((0, 4, 0), 2)]

>>> from sage.all import *
>>> R = QQ['a, b, c']; (a, b, c,) = R._first_ngens(3)
>>> f = a*c**Integer(3) + a**Integer(2)*b + Integer(2)*b**Integer(4)
>>> list(f.iterator_exp_coeff())
[((0, 4, 0), 2), ((1, 0, 3), 1), ((2, 1, 0), 1)]
>>> list(f.iterator_exp_coeff(as_ETuples=False))
[((0, 4, 0), 2), ((1, 0, 3), 1), ((2, 1, 0), 1)]

>>> R = PolynomialRing(QQ, Integer(3), order='lex', names=('a', 'b', 'c',)); (a, b, c,) = R._first_ngens(3)
>>> f = a*c**Integer(3) + a**Integer(2)*b + Integer(2)*b**Integer(4)
>>> list(f.iterator_exp_coeff())
[((2, 1, 0), 1), ((1, 0, 3), 1), ((0, 4, 0), 2)]

R.<a,b,c> = QQ[]
f = a*c^3 + a^2*b + 2*b^4
list(f.iterator_exp_coeff())
list(f.iterator_exp_coeff(as_ETuples=False))
R.<a,b,c> = PolynomialRing(QQ, 3, order='lex')
f = a*c^3 + a^2*b + 2*b^4
list(f.iterator_exp_coeff())
lc()[source]

Leading coefficient of this polynomial with respect to the term order of self.parent().

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lc()
3

sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lc()
5

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(7)), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = Integer(3)*x**Integer(1)*y**Integer(2) + Integer(2)*y**Integer(3)*z**Integer(4)
>>> f.lc()
3

>>> f = Integer(5)*x**Integer(3)*y**Integer(2)*z**Integer(4) + Integer(4)*x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lc()
5

R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
f = 3*x^1*y^2 + 2*y^3*z^4
f.lc()
f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
f.lc()
lcm(g)[source]

Return the least common multiple of self and \(g\).

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: p = (x+y)*(y+z)
sage: q = (z^4+2)*(y+z)
sage: lcm(p,q)
x*y*z^4 + y^2*z^4 + x*z^5 + y*z^5 + 2*x*y + 2*y^2 + 2*x*z + 2*y*z

sage: P.<x,y,z> = ZZ[]
sage: p = 2*(x+y)*(y+z)
sage: q = 3*(z^4+2)*(y+z)
sage: lcm(p,q)
6*x*y*z^4 + 6*y^2*z^4 + 6*x*z^5 + 6*y*z^5 + 12*x*y + 12*y^2 + 12*x*z + 12*y*z

sage: # needs sage.rings.finite_rings
sage: r.<x,y> = PolynomialRing(GF(2**8, 'a'), 2)
sage: a = r.base_ring().0
sage: f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
sage: f.lcm(x^4)
(a^2 + a)*x^6*y + (a^4 + a^3 + a)*x^4*y + (a^5)*x^4

sage: # needs sage.rings.number_field
sage: w = polygen(ZZ, 'w')
sage: r.<x,y> = PolynomialRing(NumberField(w^4 + 1, 'a'), 2)
sage: a = r.base_ring().0
sage: f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
sage: f.lcm(x^4)
(a^2 + a)*x^6*y + (a^3 + a - 1)*x^4*y + (-a)*x^4

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> p = (x+y)*(y+z)
>>> q = (z**Integer(4)+Integer(2))*(y+z)
>>> lcm(p,q)
x*y*z^4 + y^2*z^4 + x*z^5 + y*z^5 + 2*x*y + 2*y^2 + 2*x*z + 2*y*z

>>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> p = Integer(2)*(x+y)*(y+z)
>>> q = Integer(3)*(z**Integer(4)+Integer(2))*(y+z)
>>> lcm(p,q)
6*x*y*z^4 + 6*y^2*z^4 + 6*x*z^5 + 6*y*z^5 + 12*x*y + 12*y^2 + 12*x*z + 12*y*z

>>> # needs sage.rings.finite_rings
>>> r = PolynomialRing(GF(Integer(2)**Integer(8), 'a'), Integer(2), names=('x', 'y',)); (x, y,) = r._first_ngens(2)
>>> a = r.base_ring().gen(0)
>>> f = (a**Integer(2)+a)*x**Integer(2)*y + (a**Integer(4)+a**Integer(3)+a)*y + a**Integer(5)
>>> f.lcm(x**Integer(4))
(a^2 + a)*x^6*y + (a^4 + a^3 + a)*x^4*y + (a^5)*x^4

>>> # needs sage.rings.number_field
>>> w = polygen(ZZ, 'w')
>>> r = PolynomialRing(NumberField(w**Integer(4) + Integer(1), 'a'), Integer(2), names=('x', 'y',)); (x, y,) = r._first_ngens(2)
>>> a = r.base_ring().gen(0)
>>> f = (a**Integer(2)+a)*x**Integer(2)*y + (a**Integer(4)+a**Integer(3)+a)*y + a**Integer(5)
>>> f.lcm(x**Integer(4))
(a^2 + a)*x^6*y + (a^3 + a - 1)*x^4*y + (-a)*x^4

P.<x,y,z> = QQ[]
p = (x+y)*(y+z)
q = (z^4+2)*(y+z)
lcm(p,q)
P.<x,y,z> = ZZ[]
p = 2*(x+y)*(y+z)
q = 3*(z^4+2)*(y+z)
lcm(p,q)
# needs sage.rings.finite_rings
r.<x,y> = PolynomialRing(GF(2**8, 'a'), 2)
a = r.base_ring().0
f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
f.lcm(x^4)
# needs sage.rings.number_field
w = polygen(ZZ, 'w')
r.<x,y> = PolynomialRing(NumberField(w^4 + 1, 'a'), 2)
a = r.base_ring().0
f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
f.lcm(x^4)
lift(I)[source]

Given an ideal I = (f_1,...,f_r) and some g (== self) in I, find s_1,...,s_r such that g = s_1 f_1 + ... + s_r f_r.

A ValueError exception is raised if g (== self) does not belong to I.

EXAMPLES:

sage: A.<x,y> = PolynomialRing(QQ,2,order='degrevlex')
sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
sage: f = x*y^13 + y^12
sage: M = f.lift(I)
sage: M
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
sage: sum( map( mul , zip( M, I.gens() ) ) ) == f
True

>>> from sage.all import *
>>> A = PolynomialRing(QQ,Integer(2),order='degrevlex', names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> I = A.ideal([x**Integer(10) + x**Integer(9)*y**Integer(2), y**Integer(8) - x**Integer(2)*y**Integer(7) ])
>>> f = x*y**Integer(13) + y**Integer(12)
>>> M = f.lift(I)
>>> M
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
>>> sum( map( mul , zip( M, I.gens() ) ) ) == f
True

A.<x,y> = PolynomialRing(QQ,2,order='degrevlex')
I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
f = x*y^13 + y^12
M = f.lift(I)
M
sum( map( mul , zip( M, I.gens() ) ) ) == f

Check that Issue #13671 is fixed:

sage: R.<x1,x2> = QQ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: f = I.gen(0) + x2*I.gen(1)
sage: f.lift(I)
[1, x2]
sage: (f+1).lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal

>>> from sage.all import *
>>> R = QQ['x1, x2']; (x1, x2,) = R._first_ngens(2)
>>> I = R.ideal(x2**Integer(2) + x1 - Integer(2), x1**Integer(2) - Integer(1))
>>> f = I.gen(Integer(0)) + x2*I.gen(Integer(1))
>>> f.lift(I)
[1, x2]
>>> (f+Integer(1)).lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal

R.<x1,x2> = QQ[]
I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
f = I.gen(0) + x2*I.gen(1)
f.lift(I)
(f+1).lift(I)

Check that we can work over the integers:

sage: R.<x1,x2> = ZZ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: f = I.gen(0) + x2*I.gen(1)
sage: f.lift(I)
[1, x2]
sage: (f+1).lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal
sage: A.<x,y> = PolynomialRing(ZZ,2,order='degrevlex')
sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
sage: f = x*y^13 + y^12
sage: M = f.lift(I)
sage: M
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]

>>> from sage.all import *
>>> R = ZZ['x1, x2']; (x1, x2,) = R._first_ngens(2)
>>> I = R.ideal(x2**Integer(2) + x1 - Integer(2), x1**Integer(2) - Integer(1))
>>> f = I.gen(Integer(0)) + x2*I.gen(Integer(1))
>>> f.lift(I)
[1, x2]
>>> (f+Integer(1)).lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal
>>> A = PolynomialRing(ZZ,Integer(2),order='degrevlex', names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> I = A.ideal([x**Integer(10) + x**Integer(9)*y**Integer(2), y**Integer(8) - x**Integer(2)*y**Integer(7) ])
>>> f = x*y**Integer(13) + y**Integer(12)
>>> M = f.lift(I)
>>> M
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]

R.<x1,x2> = ZZ[]
I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
f = I.gen(0) + x2*I.gen(1)
f.lift(I)
(f+1).lift(I)
A.<x,y> = PolynomialRing(ZZ,2,order='degrevlex')
I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
f = x*y^13 + y^12
M = f.lift(I)
M
lm()[source]

Return the lead monomial of self with respect to the term order of self.parent(). In Sage a monomial is a product of variables in some power without a coefficient.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
sage: f = x^1*y^2 + y^3*z^4
sage: f.lm()
x*y^2
sage: f = x^3*y^2*z^4 + x^3*y^2*z^1
sage: f.lm()
x^3*y^2*z^4

sage: R.<x,y,z>=PolynomialRing(QQ, 3, order='deglex')
sage: f = x^1*y^2*z^3 + x^3*y^2*z^0
sage: f.lm()
x*y^2*z^3
sage: f = x^1*y^2*z^4 + x^1*y^1*z^5
sage: f.lm()
x*y^2*z^4

sage: R.<x,y,z>=PolynomialRing(GF(127), 3, order='degrevlex')
sage: f = x^1*y^5*z^2 + x^4*y^1*z^3
sage: f.lm()
x*y^5*z^2
sage: f = x^4*y^7*z^1 + x^4*y^2*z^3
sage: f.lm()
x^4*y^7*z

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(7)), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = x**Integer(1)*y**Integer(2) + y**Integer(3)*z**Integer(4)
>>> f.lm()
x*y^2
>>> f = x**Integer(3)*y**Integer(2)*z**Integer(4) + x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lm()
x^3*y^2*z^4

>>> R = PolynomialRing(QQ, Integer(3), order='deglex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = x**Integer(1)*y**Integer(2)*z**Integer(3) + x**Integer(3)*y**Integer(2)*z**Integer(0)
>>> f.lm()
x*y^2*z^3
>>> f = x**Integer(1)*y**Integer(2)*z**Integer(4) + x**Integer(1)*y**Integer(1)*z**Integer(5)
>>> f.lm()
x*y^2*z^4

>>> R = PolynomialRing(GF(Integer(127)), Integer(3), order='degrevlex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = x**Integer(1)*y**Integer(5)*z**Integer(2) + x**Integer(4)*y**Integer(1)*z**Integer(3)
>>> f.lm()
x*y^5*z^2
>>> f = x**Integer(4)*y**Integer(7)*z**Integer(1) + x**Integer(4)*y**Integer(2)*z**Integer(3)
>>> f.lm()
x^4*y^7*z

R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
f = x^1*y^2 + y^3*z^4
f.lm()
f = x^3*y^2*z^4 + x^3*y^2*z^1
f.lm()
R.<x,y,z>=PolynomialRing(QQ, 3, order='deglex')
f = x^1*y^2*z^3 + x^3*y^2*z^0
f.lm()
f = x^1*y^2*z^4 + x^1*y^1*z^5
f.lm()
R.<x,y,z>=PolynomialRing(GF(127), 3, order='degrevlex')
f = x^1*y^5*z^2 + x^4*y^1*z^3
f.lm()
f = x^4*y^7*z^1 + x^4*y^2*z^3
f.lm()
local_height(v, prec=None)[source]

Return the maximum of the local height of the coefficients of this polynomial.

INPUT:

  • v – a prime or prime ideal of the base ring

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 1/1331*x^2 + 1/4000*y^2
sage: f.local_height(1331)
7.19368581839511

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(1331)*x**Integer(2) + Integer(1)/Integer(4000)*y**Integer(2)
>>> f.local_height(Integer(1331))
7.19368581839511

R.<x,y> = PolynomialRing(QQ)
f = 1/1331*x^2 + 1/4000*y^2
f.local_height(1331)

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 - 5)
sage: T.<t,w> = K[]
sage: I = K.ideal(3)
sage: f = 1/3*t*w + 3
sage: f.local_height(I)
1.09861228866811

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) - Integer(5), names=('k',)); (k,) = K._first_ngens(1)
>>> T = K['t, w']; (t, w,) = T._first_ngens(2)
>>> I = K.ideal(Integer(3))
>>> f = Integer(1)/Integer(3)*t*w + Integer(3)
>>> f.local_height(I)
1.09861228866811

# needs sage.rings.number_field
R.<x> = QQ[]
K.<k> = NumberField(x^2 - 5)
T.<t,w> = K[]
I = K.ideal(3)
f = 1/3*t*w + 3
f.local_height(I)

sage: R.<x,y> = QQ[]
sage: f = 1/2*x*y + 2
sage: f.local_height(2, prec=2)
0.75

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(2)*x*y + Integer(2)
>>> f.local_height(Integer(2), prec=Integer(2))
0.75

R.<x,y> = QQ[]
f = 1/2*x*y + 2
f.local_height(2, prec=2)
local_height_arch(i, prec=None)[source]

Return the maximum of the local height at the i-th infinite place of the coefficients of this polynomial.

INPUT:

  • i – integer

  • prec – desired floating point precision (default: default RealField precision)

OUTPUT: a real number

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = 210*x*y
sage: f.local_height_arch(0)
5.34710753071747

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = Integer(210)*x*y
>>> f.local_height_arch(Integer(0))
5.34710753071747

R.<x,y> = PolynomialRing(QQ)
f = 210*x*y
f.local_height_arch(0)

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<k> = NumberField(x^2 - 5)
sage: T.<t,w> = K[]
sage: f = 1/2*t*w + 3
sage: f.local_height_arch(1, prec=52)
1.09861228866811

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(2) - Integer(5), names=('k',)); (k,) = K._first_ngens(1)
>>> T = K['t, w']; (t, w,) = T._first_ngens(2)
>>> f = Integer(1)/Integer(2)*t*w + Integer(3)
>>> f.local_height_arch(Integer(1), prec=Integer(52))
1.09861228866811

# needs sage.rings.number_field
R.<x> = QQ[]
K.<k> = NumberField(x^2 - 5)
T.<t,w> = K[]
f = 1/2*t*w + 3
f.local_height_arch(1, prec=52)

sage: R.<x,y> = QQ[]
sage: f = 1/2*x*y + 3
sage: f.local_height_arch(0, prec=2)
1.0

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(1)/Integer(2)*x*y + Integer(3)
>>> f.local_height_arch(Integer(0), prec=Integer(2))
1.0

R.<x,y> = QQ[]
f = 1/2*x*y + 3
f.local_height_arch(0, prec=2)
lt()[source]

Leading term of this polynomial. In Sage a term is a product of variables in some power and a coefficient.

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lt()
3*x*y^2

sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lt()
-2*x^3*y^2*z^4

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(7)), Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = Integer(3)*x**Integer(1)*y**Integer(2) + Integer(2)*y**Integer(3)*z**Integer(4)
>>> f.lt()
3*x*y^2

>>> f = Integer(5)*x**Integer(3)*y**Integer(2)*z**Integer(4) + Integer(4)*x**Integer(3)*y**Integer(2)*z**Integer(1)
>>> f.lt()
-2*x^3*y^2*z^4

R.<x,y,z> = PolynomialRing(GF(7), 3, order='lex')
f = 3*x^1*y^2 + 2*y^3*z^4
f.lt()
f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
f.lt()
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

OUTPUT: coefficient in base ring

See also

For coefficients in a base ring of fewer variables, look at coefficient.

EXAMPLES:

sage: P.<x,y> = QQ[]

The parent of the return is a member of the base ring.
sage: f = 2 * x * y
sage: c = f.monomial_coefficient(x*y); c
2
sage: c.parent()
Rational Field

sage: f = y^2 + y^2*x - x^9 - 7*x + 5*x*y
sage: f.monomial_coefficient(y^2)
1
sage: f.monomial_coefficient(x*y)
5
sage: f.monomial_coefficient(x^9)
-1
sage: f.monomial_coefficient(x^10)
0

>>> from sage.all import *
>>> P = QQ['x, y']; (x, y,) = P._first_ngens(2)

The parent of the return is a member of the base ring.
>>> f = Integer(2) * x * y
>>> c = f.monomial_coefficient(x*y); c
2
>>> c.parent()
Rational Field

>>> f = y**Integer(2) + y**Integer(2)*x - x**Integer(9) - Integer(7)*x + Integer(5)*x*y
>>> f.monomial_coefficient(y**Integer(2))
1
>>> f.monomial_coefficient(x*y)
5
>>> f.monomial_coefficient(x**Integer(9))
-1
>>> f.monomial_coefficient(x**Integer(10))
0

P.<x,y> = QQ[]
f = 2 * x * y
c = f.monomial_coefficient(x*y); c
c.parent()
f = y^2 + y^2*x - x^9 - 7*x + 5*x*y
f.monomial_coefficient(y^2)
f.monomial_coefficient(x*y)
f.monomial_coefficient(x^9)
f.monomial_coefficient(x^10)
monomial_coefficients()[source]

Return a dictionary representing self. This dictionary is in the same format as the generic MPolynomial: The dictionary consists of ETuple:coefficient pairs.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: f = 2*x*y^3*z^2 + 1/7*x^2 + 2/3
sage: f.monomial_coefficients()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> f = Integer(2)*x*y**Integer(3)*z**Integer(2) + Integer(1)/Integer(7)*x**Integer(2) + Integer(2)/Integer(3)
>>> f.monomial_coefficients()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

R.<x,y,z> = QQ[]
f = 2*x*y^3*z^2 + 1/7*x^2 + 2/3
f.monomial_coefficients()

dict is an alias:

sage: f.dict()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

>>> from sage.all import *
>>> f.dict()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}

f.dict()
monomials()[source]

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

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: f = x + 3/2*y*z^2 + 2/3
sage: f.monomials()
[y*z^2, x, 1]
sage: f = P(3/2)
sage: f.monomials()
[1]

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = x + Integer(3)/Integer(2)*y*z**Integer(2) + Integer(2)/Integer(3)
>>> f.monomials()
[y*z^2, x, 1]
>>> f = P(Integer(3)/Integer(2))
>>> f.monomials()
[1]

P.<x,y,z> = QQ[]
f = x + 3/2*y*z^2 + 2/3
f.monomials()
f = P(3/2)
f.monomials()
number_of_terms()[source]

Return the number of nonzero coefficients of this polynomial.

This is also called weight, hamming_weight() or sparsity.

EXAMPLES:

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

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(3) - y
>>> f.number_of_terms()
2
>>> R(Integer(0)).number_of_terms()
0
>>> f = (x+y)**Integer(100)
>>> f.number_of_terms()
101

R.<x, y> = ZZ[]
f = x^3 - y
f.number_of_terms()
R(0).number_of_terms()
f = (x+y)^100
f.number_of_terms()

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

>>> from sage.all import *
>>> f.hamming_weight()
101

f.hamming_weight()
numerator()[source]

Return a numerator of self computed as self * self.denominator().

If the base_field of self is the Rational Field then the numerator is a polynomial whose base_ring is the Integer Ring, this is done for compatibility to the univariate case.

Warning

This is not the numerator of the rational function defined by self, which would always be self since self is a polynomial.

EXAMPLES:

First we compute the numerator of a polynomial with integer coefficients, which is of course self.

sage: R.<x, y> = ZZ[]
sage: f = x^3 + 17*y + 1
sage: f.numerator()
x^3 + 17*y + 1
sage: f == f.numerator()
True

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(3) + Integer(17)*y + Integer(1)
>>> f.numerator()
x^3 + 17*y + 1
>>> f == f.numerator()
True

R.<x, y> = ZZ[]
f = x^3 + 17*y + 1
f.numerator()
f == f.numerator()

Next we compute the numerator of a polynomial with rational coefficients.

sage: R.<x,y> = PolynomialRing(QQ)
sage: f = (1/17)*x^19 - (2/3)*y + 1/3; f
1/17*x^19 - 2/3*y + 1/3
sage: f.numerator()
3*x^19 - 34*y + 17
sage: f == f.numerator()
False
sage: f.numerator().base_ring()
Integer Ring

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> f = (Integer(1)/Integer(17))*x**Integer(19) - (Integer(2)/Integer(3))*y + Integer(1)/Integer(3); f
1/17*x^19 - 2/3*y + 1/3
>>> f.numerator()
3*x^19 - 34*y + 17
>>> f == f.numerator()
False
>>> f.numerator().base_ring()
Integer Ring

R.<x,y> = PolynomialRing(QQ)
f = (1/17)*x^19 - (2/3)*y + 1/3; f
f.numerator()
f == f.numerator()
f.numerator().base_ring()

We check that the computation of numerator and denominator is valid.

sage: K=QQ['x,y']
sage: f=K.random_element()
sage: f.numerator() / f.denominator() == f
True

>>> from sage.all import *
>>> K=QQ['x,y']
>>> f=K.random_element()
>>> f.numerator() / f.denominator() == f
True

K=QQ['x,y']
f=K.random_element()
f.numerator() / f.denominator() == f

The following tests against a bug fixed in Issue #11780:

sage: P.<foo,bar> = ZZ[]
sage: Q.<foo,bar> = QQ[]
sage: f = Q.random_element()
sage: f.numerator().parent() is P
True

>>> from sage.all import *
>>> P = ZZ['foo, bar']; (foo, bar,) = P._first_ngens(2)
>>> Q = QQ['foo, bar']; (foo, bar,) = Q._first_ngens(2)
>>> f = Q.random_element()
>>> f.numerator().parent() is P
True

P.<foo,bar> = ZZ[]
Q.<foo,bar> = QQ[]
f = Q.random_element()
f.numerator().parent() is P
nvariables()[source]

Return the number variables in this polynomial.

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(GF(127))
sage: f = x*y + z
sage: f.nvariables()
3
sage: f = x + y
sage: f.nvariables()
2

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(127)), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> f = x*y + z
>>> f.nvariables()
3
>>> f = x + y
>>> f.nvariables()
2

P.<x,y,z> = PolynomialRing(GF(127))
f = x*y + z
f.nvariables()
f = x + y
f.nvariables()
quo_rem(right)[source]

Return quotient and remainder of self and right.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = y*x^2 + x + 1
sage: f.quo_rem(x)
(x*y + 1, 1)
sage: f.quo_rem(y)
(x^2, x + 1)

sage: R.<x,y> = ZZ[]
sage: f = 2*y*x^2 + x + 1
sage: f.quo_rem(x)
(2*x*y + 1, 1)
sage: f.quo_rem(y)
(2*x^2, x + 1)
sage: f.quo_rem(3*x)
(0, 2*x^2*y + x + 1)

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = y*x**Integer(2) + x + Integer(1)
>>> f.quo_rem(x)
(x*y + 1, 1)
>>> f.quo_rem(y)
(x^2, x + 1)

>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(2)*y*x**Integer(2) + x + Integer(1)
>>> f.quo_rem(x)
(2*x*y + 1, 1)
>>> f.quo_rem(y)
(2*x^2, x + 1)
>>> f.quo_rem(Integer(3)*x)
(0, 2*x^2*y + x + 1)

R.<x,y> = QQ[]
f = y*x^2 + x + 1
f.quo_rem(x)
f.quo_rem(y)
R.<x,y> = ZZ[]
f = 2*y*x^2 + x + 1
f.quo_rem(x)
f.quo_rem(y)
f.quo_rem(3*x)
reduce(I)[source]

Return a remainder of this polynomial modulo the polynomials in I.

INPUT:

  • I – an ideal or a list/set/iterable of polynomials

OUTPUT: a polynomial r such that:

  • self – r is in the ideal generated by I.

  • No term in r is divisible by any of the leading monomials of I.

The result r is canonical if:

  • I is an ideal, and Sage can compute a Groebner basis of it.

  • I is a list/set/iterable that is a (strong) Groebner basis for the term order of self. (A strong Groebner basis is such that for every leading term t of the ideal generated by I, there exists an element g of I such that the leading term of g divides t.)

The result r is implementation-dependent (and possibly order-dependent) otherwise. If I is an ideal and no Groebner basis can be computed, its list of generators I.gens() is used for the reduction.

EXAMPLES:

sage: P.<x,y,z> = QQ[]
sage: f1 = -2 * x^2 + x^3
sage: f2 = -2 * y + x* y
sage: f3 = -x^2 + y^2
sage: F = Ideal([f1,f2,f3])
sage: g = x*y - 3*x*y^2
sage: g.reduce(F)
-6*y^2 + 2*y
sage: g.reduce(F.gens())
-6*y^2 + 2*y

>>> from sage.all import *
>>> P = QQ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f1 = -Integer(2) * x**Integer(2) + x**Integer(3)
>>> f2 = -Integer(2) * y + x* y
>>> f3 = -x**Integer(2) + y**Integer(2)
>>> F = Ideal([f1,f2,f3])
>>> g = x*y - Integer(3)*x*y**Integer(2)
>>> g.reduce(F)
-6*y^2 + 2*y
>>> g.reduce(F.gens())
-6*y^2 + 2*y

P.<x,y,z> = QQ[]
f1 = -2 * x^2 + x^3
f2 = -2 * y + x* y
f3 = -x^2 + y^2
F = Ideal([f1,f2,f3])
g = x*y - 3*x*y^2
g.reduce(F)
g.reduce(F.gens())

\(\ZZ\) is also supported.

sage: P.<x,y,z> = ZZ[]
sage: f1 = -2 * x^2 + x^3
sage: f2 = -2 * y + x* y
sage: f3 = -x^2 + y^2
sage: F = Ideal([f1,f2,f3])
sage: g = x*y - 3*x*y^2
sage: g.reduce(F)
-6*y^2 + 2*y
sage: g.reduce(F.gens())
-6*y^2 + 2*y

sage: f = 3*x
sage: f.reduce([2*x,y])
x

>>> from sage.all import *
>>> P = ZZ['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f1 = -Integer(2) * x**Integer(2) + x**Integer(3)
>>> f2 = -Integer(2) * y + x* y
>>> f3 = -x**Integer(2) + y**Integer(2)
>>> F = Ideal([f1,f2,f3])
>>> g = x*y - Integer(3)*x*y**Integer(2)
>>> g.reduce(F)
-6*y^2 + 2*y
>>> g.reduce(F.gens())
-6*y^2 + 2*y

>>> f = Integer(3)*x
>>> f.reduce([Integer(2)*x,y])
x

P.<x,y,z> = ZZ[]
f1 = -2 * x^2 + x^3
f2 = -2 * y + x* y
f3 = -x^2 + y^2
F = Ideal([f1,f2,f3])
g = x*y - 3*x*y^2
g.reduce(F)
g.reduce(F.gens())
f = 3*x
f.reduce([2*x,y])

The reduction is not canonical when I is not a Groebner basis:

sage: A.<x,y> = QQ[]
sage: (x+y).reduce([x+y, x-y])
2*y
sage: (x+y).reduce([x-y, x+y])
0

>>> from sage.all import *
>>> A = QQ['x, y']; (x, y,) = A._first_ngens(2)
>>> (x+y).reduce([x+y, x-y])
2*y
>>> (x+y).reduce([x-y, x+y])
0

A.<x,y> = QQ[]
(x+y).reduce([x+y, x-y])
(x+y).reduce([x-y, x+y])
resultant(other, variable=None)[source]

Compute the resultant of this polynomial and the first argument with respect to the variable given as the second argument.

If a second argument is not provide the first variable of the parent is chosen.

INPUT:

  • other – polynomial

  • variable – optional variable (default: None)

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: a = x+y
sage: b = x^3-y^3
sage: c = a.resultant(b); c
-2*y^3
sage: d = a.resultant(b,y); d
2*x^3

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> a = x+y
>>> b = x**Integer(3)-y**Integer(3)
>>> c = a.resultant(b); c
-2*y^3
>>> d = a.resultant(b,y); d
2*x^3

P.<x,y> = PolynomialRing(QQ,2)
a = x+y
b = x^3-y^3
c = a.resultant(b); c
d = a.resultant(b,y); d

The SINGULAR example:

sage: R.<x,y,z> = PolynomialRing(GF(32003), 3)                              # needs sage.rings.finite_rings
sage: f = 3 * (x+2)^3 + y
sage: g = x + y + z                                                         # needs sage.rings.finite_rings
sage: f.resultant(g, x)                                                     # needs sage.rings.finite_rings
3*y^3 + 9*y^2*z + 9*y*z^2 + 3*z^3 - 18*y^2 - 36*y*z - 18*z^2 + 35*y + 36*z - 24

>>> from sage.all import *
>>> R = PolynomialRing(GF(Integer(32003)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)# needs sage.rings.finite_rings
>>> f = Integer(3) * (x+Integer(2))**Integer(3) + y
>>> g = x + y + z                                                         # needs sage.rings.finite_rings
>>> f.resultant(g, x)                                                     # needs sage.rings.finite_rings
3*y^3 + 9*y^2*z + 9*y*z^2 + 3*z^3 - 18*y^2 - 36*y*z - 18*z^2 + 35*y + 36*z - 24

R.<x,y,z> = PolynomialRing(GF(32003), 3)                              # needs sage.rings.finite_rings
f = 3 * (x+2)^3 + y
g = x + y + z                                                         # needs sage.rings.finite_rings
f.resultant(g, x)                                                     # needs sage.rings.finite_rings

Resultants are also supported over the Integers:

sage: R.<x,y,a,b,u> = PolynomialRing(ZZ, 5, order='lex')
sage: r = (x^4*y^2 + x^2*y - y).resultant(x*y - y*a - x*b + a*b + u, x)
sage: r
y^6*a^4 - 4*y^5*a^4*b - 4*y^5*a^3*u + y^5*a^2 - y^5 + 6*y^4*a^4*b^2 + 12*y^4*a^3*b*u - 4*y^4*a^2*b + 6*y^4*a^2*u^2 - 2*y^4*a*u + 4*y^4*b - 4*y^3*a^4*b^3 - 12*y^3*a^3*b^2*u + 6*y^3*a^2*b^2 - 12*y^3*a^2*b*u^2 + 6*y^3*a*b*u - 4*y^3*a*u^3 - 6*y^3*b^2 + y^3*u^2 + y^2*a^4*b^4 + 4*y^2*a^3*b^3*u - 4*y^2*a^2*b^3 + 6*y^2*a^2*b^2*u^2 - 6*y^2*a*b^2*u + 4*y^2*a*b*u^3 + 4*y^2*b^3 - 2*y^2*b*u^2 + y^2*u^4 + y*a^2*b^4 + 2*y*a*b^3*u - y*b^4 + y*b^2*u^2

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, Integer(5), order='lex', names=('x', 'y', 'a', 'b', 'u',)); (x, y, a, b, u,) = R._first_ngens(5)
>>> r = (x**Integer(4)*y**Integer(2) + x**Integer(2)*y - y).resultant(x*y - y*a - x*b + a*b + u, x)
>>> r
y^6*a^4 - 4*y^5*a^4*b - 4*y^5*a^3*u + y^5*a^2 - y^5 + 6*y^4*a^4*b^2 + 12*y^4*a^3*b*u - 4*y^4*a^2*b + 6*y^4*a^2*u^2 - 2*y^4*a*u + 4*y^4*b - 4*y^3*a^4*b^3 - 12*y^3*a^3*b^2*u + 6*y^3*a^2*b^2 - 12*y^3*a^2*b*u^2 + 6*y^3*a*b*u - 4*y^3*a*u^3 - 6*y^3*b^2 + y^3*u^2 + y^2*a^4*b^4 + 4*y^2*a^3*b^3*u - 4*y^2*a^2*b^3 + 6*y^2*a^2*b^2*u^2 - 6*y^2*a*b^2*u + 4*y^2*a*b*u^3 + 4*y^2*b^3 - 2*y^2*b*u^2 + y^2*u^4 + y*a^2*b^4 + 2*y*a*b^3*u - y*b^4 + y*b^2*u^2

R.<x,y,a,b,u> = PolynomialRing(ZZ, 5, order='lex')
r = (x^4*y^2 + x^2*y - y).resultant(x*y - y*a - x*b + a*b + u, x)
r
sub_m_mul_q(m, q)[source]

Return self - m*q, where m must be a monomial and q a polynomial.

INPUT:

  • m – a monomial

  • q – a polynomial

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: x.sub_m_mul_q(y, z)
-y*z + x

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> x.sub_m_mul_q(y, z)
-y*z + x

P.<x,y,z> = PolynomialRing(QQ,3)
x.sub_m_mul_q(y, z)
subs(fixed=None, **kw)[source]

Fixes some given variables in a given multivariate polynomial and returns the changed multivariate polynomials. The polynomial itself is not affected. The variable,value pairs for fixing are to be provided as dictionary of the form {variable:value}.

This is a special case of evaluating the polynomial with some of the variables constants and the others the original variables, but should be much faster if only few variables are to be fixed.

INPUT:

  • fixed – (optional) dict with variable:value pairs

  • **kw – names parameters

OUTPUT: a new multivariate polynomial

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = x^2 + y + x^2*y^2 + 5
sage: f(5,y)
25*y^2 + y + 30
sage: f.subs({x: 5})
25*y^2 + y + 30
sage: f.subs(x=5)
25*y^2 + y + 30

sage: P.<x,y,z> = PolynomialRing(GF(2), 3)
sage: f = x + y + 1
sage: f.subs({x:y+1})
0
sage: f.subs(x=y)
1
sage: f.subs(x=x)
x + y + 1
sage: f.subs({x: z})
y + z + 1
sage: f.subs(x=z + 1)
y + z

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

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(2) + y + x**Integer(2)*y**Integer(2) + Integer(5)
>>> f(Integer(5),y)
25*y^2 + y + 30
>>> f.subs({x: Integer(5)})
25*y^2 + y + 30
>>> f.subs(x=Integer(5))
25*y^2 + y + 30

>>> P = PolynomialRing(GF(Integer(2)), Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> f = x + y + Integer(1)
>>> f.subs({x:y+Integer(1)})
0
>>> f.subs(x=y)
1
>>> f.subs(x=x)
x + y + 1
>>> f.subs({x: z})
y + z + 1
>>> f.subs(x=z + Integer(1))
y + z

>>> f.subs(x=Integer(1)/y)
(y^2 + y + 1)/y
>>> f.subs({x: Integer(1)/y})
(y^2 + y + 1)/y

R.<x,y> = QQ[]
f = x^2 + y + x^2*y^2 + 5
f(5,y)
f.subs({x: 5})
f.subs(x=5)
P.<x,y,z> = PolynomialRing(GF(2), 3)
f = x + y + 1
f.subs({x:y+1})
f.subs(x=y)
f.subs(x=x)
f.subs({x: z})
f.subs(x=z + 1)
f.subs(x=1/y)
f.subs({x: 1/y})

The parameters are substituted in order and without side effects:

sage: R.<x,y>=QQ[]
sage: g=x+y
sage: g.subs({x:x+1,y:x*y})
x*y + x + 1
sage: g.subs({x:x+1}).subs({y:x*y})
x*y + x + 1
sage: g.subs({y:x*y}).subs({x:x+1})
x*y + x + y + 1

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> g=x+y
>>> g.subs({x:x+Integer(1),y:x*y})
x*y + x + 1
>>> g.subs({x:x+Integer(1)}).subs({y:x*y})
x*y + x + 1
>>> g.subs({y:x*y}).subs({x:x+Integer(1)})
x*y + x + y + 1

R.<x,y>=QQ[]
g=x+y
g.subs({x:x+1,y:x*y})
g.subs({x:x+1}).subs({y:x*y})
g.subs({y:x*y}).subs({x:x+1})

sage: R.<x,y> = QQ[]
sage: f = x + 2*y
sage: f.subs(x=y,y=x)
2*x + y

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x + Integer(2)*y
>>> f.subs(x=y,y=x)
2*x + y

R.<x,y> = QQ[]
f = x + 2*y
f.subs(x=y,y=x)
total_degree(std_grading=False)[source]

Return the total degree of self, which is the maximum degree of all monomials in self.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: f = 2*x*y^3*z^2
sage: f.total_degree()
6
sage: f = 4*x^2*y^2*z^3
sage: f.total_degree()
7
sage: f = 99*x^6*y^3*z^9
sage: f.total_degree()
18
sage: f = x*y^3*z^6+3*x^2
sage: f.total_degree()
10
sage: f = z^3+8*x^4*y^5*z
sage: f.total_degree()
10
sage: f = z^9+10*x^4+y^8*x^2
sage: f.total_degree()
10

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> f = Integer(2)*x*y**Integer(3)*z**Integer(2)
>>> f.total_degree()
6
>>> f = Integer(4)*x**Integer(2)*y**Integer(2)*z**Integer(3)
>>> f.total_degree()
7
>>> f = Integer(99)*x**Integer(6)*y**Integer(3)*z**Integer(9)
>>> f.total_degree()
18
>>> f = x*y**Integer(3)*z**Integer(6)+Integer(3)*x**Integer(2)
>>> f.total_degree()
10
>>> f = z**Integer(3)+Integer(8)*x**Integer(4)*y**Integer(5)*z
>>> f.total_degree()
10
>>> f = z**Integer(9)+Integer(10)*x**Integer(4)+y**Integer(8)*x**Integer(2)
>>> f.total_degree()
10

R.<x,y,z> = QQ[]
f = 2*x*y^3*z^2
f.total_degree()
f = 4*x^2*y^2*z^3
f.total_degree()
f = 99*x^6*y^3*z^9
f.total_degree()
f = x*y^3*z^6+3*x^2
f.total_degree()
f = z^3+8*x^4*y^5*z
f.total_degree()
f = z^9+10*x^4+y^8*x^2
f.total_degree()

A matrix term ordering changes the grading. To get the total degree using the standard grading, use std_grading=True:

sage: tord = TermOrder(matrix(3, [3,2,1,1,1,0,1,0,0]))
sage: tord
Matrix term order with matrix
[3 2 1]
[1 1 0]
[1 0 0]
sage: R.<x,y,z> = PolynomialRing(QQ, order=tord)
sage: f = x^2*y
sage: f.total_degree()
8
sage: f.total_degree(std_grading=True)
3

>>> from sage.all import *
>>> tord = TermOrder(matrix(Integer(3), [Integer(3),Integer(2),Integer(1),Integer(1),Integer(1),Integer(0),Integer(1),Integer(0),Integer(0)]))
>>> tord
Matrix term order with matrix
[3 2 1]
[1 1 0]
[1 0 0]
>>> R = PolynomialRing(QQ, order=tord, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> f = x**Integer(2)*y
>>> f.total_degree()
8
>>> f.total_degree(std_grading=True)
3

tord = TermOrder(matrix(3, [3,2,1,1,1,0,1,0,0]))
tord
R.<x,y,z> = PolynomialRing(QQ, order=tord)
f = x^2*y
f.total_degree()
f.total_degree(std_grading=True)
univariate_polynomial(R=None)[source]

Return a univariate polynomial associated to this multivariate polynomial.

INPUT:

  • R – (default: None) PolynomialRing

If this polynomial is not in at most one variable, then a ValueError exception is raised. This is checked using the is_univariate() method. The new Polynomial is over the same base ring as the given MPolynomial and in the variable x if no ring R is provided.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: f = 3*x^2 - 2*y + 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.subs({x:10}); g
700*y^2 - 2*y + 305
sage: g.univariate_polynomial ()
700*y^2 - 2*y + 305
sage: g.univariate_polynomial(PolynomialRing(QQ,'z'))
700*z^2 - 2*z + 305

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = Integer(3)*x**Integer(2) - Integer(2)*y + Integer(7)*x**Integer(2)*y**Integer(2) + Integer(5)
>>> f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
>>> g = f.subs({x:Integer(10)}); g
700*y^2 - 2*y + 305
>>> g.univariate_polynomial ()
700*y^2 - 2*y + 305
>>> g.univariate_polynomial(PolynomialRing(QQ,'z'))
700*z^2 - 2*z + 305

R.<x, y> = QQ[]
f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
f.univariate_polynomial()
g = f.subs({x:10}); g
g.univariate_polynomial ()
g.univariate_polynomial(PolynomialRing(QQ,'z'))

Here’s an example with a constant multivariate polynomial:

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

>>> from sage.all import *
>>> g = R(Integer(1))
>>> h = g.univariate_polynomial(); h
1
>>> h.parent()
Univariate Polynomial Ring in x over Rational Field

g = R(1)
h = g.univariate_polynomial(); h
h.parent()
variable(i=0)[source]

Return the \(i\)-th variable occurring in self. The index \(i\) is the index in self.variables().

EXAMPLES:

sage: P.<x,y,z> = GF(2)[]
sage: f = x*z^2 + z + 1
sage: f.variables()
(x, z)
sage: f.variable(1)
z

>>> from sage.all import *
>>> P = GF(Integer(2))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = x*z**Integer(2) + z + Integer(1)
>>> f.variables()
(x, z)
>>> f.variable(Integer(1))
z

P.<x,y,z> = GF(2)[]
f = x*z^2 + z + 1
f.variables()
f.variable(1)
variables()[source]

Return a tuple of all variables occurring in self.

EXAMPLES:

sage: P.<x,y,z> = GF(2)[]
sage: f = x*z^2 + z + 1
sage: f.variables()
(x, z)

>>> from sage.all import *
>>> P = GF(Integer(2))['x, y, z']; (x, y, z,) = P._first_ngens(3)
>>> f = x*z**Integer(2) + z + Integer(1)
>>> f.variables()
(x, z)

P.<x,y,z> = GF(2)[]
f = x*z^2 + z + 1
f.variables()
sage.rings.polynomial.multi_polynomial_libsingular.unpickle_MPolynomialRing_libsingular(base_ring, names, term_order)[source]

Inverse function for MPolynomialRing_libsingular.__reduce__.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ)
sage: loads(dumps(P)) is P # indirect doctest
True

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> loads(dumps(P)) is P # indirect doctest
True

P.<x,y> = PolynomialRing(QQ)
loads(dumps(P)) is P # indirect doctest
sage.rings.polynomial.multi_polynomial_libsingular.unpickle_MPolynomial_libsingular(R, d)[source]

Deserialize an MPolynomial_libsingular object.

INPUT:

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ)
sage: loads(dumps(x)) == x # indirect doctest
True

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> loads(dumps(x)) == x # indirect doctest
True

P.<x,y> = PolynomialRing(QQ)
loads(dumps(x)) == x # indirect doctest