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')
sage: 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)
sage: P.<x,y> = PolynomialRing(K, 2)
sage: 1/2*s*x^2 + 3/4*s
(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)
<type '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'>

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)
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
sage: (x^5 + 2*y^5)^3
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15

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

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

Bases: sage.rings.polynomial.multi_polynomial_ring_base.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[]
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')
sage: 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)
sage: P.<x,y> = PolynomialRing(K, 2)
sage: 1/2*s*x^2 + 3/4*s
(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)
<type '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

Element

alias of MPolynomial_libsingular

gen(n=0)

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

INPUT:

• n – an integer >= 0

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

ideal(*gens, **kwds)

Create an ideal in this polynomial ring.

INPUT:

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

• coerce - bool (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)
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

monomial_all_divisors(t)

Return a list of all monomials that divide t.

Coefficients are ignored.

INPUT:

• t - a monomial

OUTPUT:

a 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]

ALGORITHM: addwithcarry idea by Toon Segers

monomial_divides(a, b)

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

monomial_lcm(f, g)

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

monomial_pairwise_prime(g, h)

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

monomial_quotient(f, g, coeff=False)

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

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.

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)

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)

ngens()

Returns 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)
sage: P = PolynomialRing(k,1000,'x')
sage: P.ngens()
1000

class sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular

Bases: sage.rings.polynomial.multi_polynomial.MPolynomial

A multivariate polynomial implemented using libSINGULAR.

add_m_mul_q(m, q)

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

coefficient(degrees)

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

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

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

AUTHOR:

• Joel B. Mohler (2007.10.31)

coefficients()

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]

AUTHOR:

• Didier Deshommes

constant_coefficient()

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

degree(x=None, std_grading=False)

Return the degree of this polynomial.

INPUT:

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

OUTPUT:

If x is not given, return the maximum degree of the monomials of the polynomial. Note that the degree of a monomial is affected by the gradings given to the generators of the parent ring. If x is given, it is (or coercible to) a generator of the parent ring and the output is the maximum degree in x. This is not affected by the gradings 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

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]

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

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

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

degrees()

Returns 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)

dict()

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

divides(other)

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

exponents(as_ETuples=True)

Return the exponents of the monomials appearing in this polynomial.

INPUT:

• as_ETuples – (default: True) if True returns the result as an 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)]

factor(proof=None)

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()
sage: F
x * (x^2 + x + 1) * (x^2 + 2*y^2)

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

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

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()
sage: F
((-a)) * x * (x - 1)^2 * ((-a + 1)*x^2 + y^2)
sage: f - F
0

Next we factor a polynomial over a number field.:

sage: p = var('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

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

sage: x = var('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

Another example:

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

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

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

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

Factorization over the integers is now supported, see trac ticket #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)

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.

gcd(right, algorithm=None, **kwds)

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

We compute a gcd over a finite field:

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

We compute a gcd over a 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

gradient()

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]

hamming_weight()

Return the number of non-zero 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

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

integral(var)

Integrates 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

Check that trac ticket #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

inverse_of_unit()

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

is_constant()

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

is_homogeneous()

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

is_monomial()

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

is_squarefree()

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

is_term()

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

is_univariate()

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

is_zero()

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

iterator_exp_coeff(as_ETuples=True)

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

INPUT:

• as_ETuples – (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)]

lc()

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

lcm(g)

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: 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: w = var('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

lift(I)

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

Check that trac ticket #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
sage: f.lift(I)
[1, x2]

lm()

Returns 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

lt()

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

monomial_coefficient(mon)

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

monomials()

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]

number_of_terms()

Return the number of non-zero 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

The method hamming_weight() is an alias:

sage: f.hamming_weight()
101

numerator()

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

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

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

The following tests against a bug fixed in trac ticket #11780:

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

nvariables()

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

quo_rem(right)

Returns 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)

reduce(I)

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

\(\ZZ\) 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])
3*x

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

resultant(other, variable=None)

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

The SINGULAR example:

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

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

sub_m_mul_q(m, q)

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

subs(fixed=None, **kw)

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

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

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

total_degree(std_grading=False)

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

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

univariate_polynomial(R=None)

Returns 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

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

variable(i=0)

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

variables()

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)

sage.rings.polynomial.multi_polynomial_libsingular.unpickle_MPolynomialRing_libsingular(base_ring, names, term_order)

inverse function for MPolynomialRing_libsingular.__reduce__

EXAMPLES:

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

sage.rings.polynomial.multi_polynomial_libsingular.unpickle_MPolynomial_libsingular(R, d)

Deserialize an MPolynomial_libsingular object

INPUT:

• R - the base ring

• d - a Python dictionary as returned by MPolynomial_libsingular.dict()

EXAMPLES:

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