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'>
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
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
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_baseConstruct 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/tupleorder– 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
- Element[source]#
alias of
MPolynomial_libsingular
- gen(n=0)[source]#
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
>>> 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
- ideal(*gens, **kwds)[source]#
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 toFalseif 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
- monomial_all_divisors(t)[source]#
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]
>>> 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]
ALGORITHM: addwithcarry idea by Toon Segers
- monomial_divides(a, b)[source]#
Return
Falseif a does not divide b andTrueotherwise.Coefficients are ignored.
INPUT:
a– monomialb– 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
- monomial_lcm(f, g)[source]#
LCM for monomials. Coefficients are ignored.
INPUT:
f– monomialg– 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
- monomial_pairwise_prime(g, h)[source]#
Return
Trueifhandgare pairwise prime. Both are treated as monomials.Coefficients are ignored.
INPUT:
h– monomialg– 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
- monomial_quotient(f, g, coeff=False)[source]#
Return
f/g, where bothfand``gare treated as monomials.Coefficients are ignored by default.
INPUT:
f– monomialg– monomialcoeff– 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
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.
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
ginGwhereg.lm()dividesf. If found(flt,g)is returned,(0,0)otherwise, wherefltisf/g.lm().It is assumed that
Gis iterable and contains only elements in this polynomial ring.Coefficients are ignored.
INPUT:
f– monomialG– 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)
- ngens()[source]#
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) # 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
- class sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular[source]#
Bases:
MPolynomial_libsingularA multivariate polynomial implemented using libSINGULAR.
- add_m_mul_q(m, q)[source]#
Return
self + m*q, wheremmust be a monomial andqa polynomial.INPUT:
m– a monomialq– 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
- 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 indegrees. However, the result has the same parent as this polynomial.This function contrasts with the function
monomial_coefficientwhich 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 (withNonein 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
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
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
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]
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
- 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
xisNone, return the total degree ofself. Note that this result is affected by the weighting given to the generators of the parent ring. Otherwise, ifxis (or is coercible to) a generator of the parent ring, the output is the maximum degree ofxinself. 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
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
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]
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
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
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
- degrees()[source]#
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)
>>> 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)
- dict()[source]#
Return a dictionary representing self. This dictionary is in the same format as the generic MPolynomial: The dictionary consists of
ETuple:coefficientpairs.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}
>>> 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.dict() {(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}
- divides(other)[source]#
Return
Trueif this polynomial dividesother.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
- exponents(as_ETuples=True)[source]#
Return the exponents of the monomials appearing in this polynomial.
INPUT:
as_ETuples– (default:True) ifTruereturns 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)]
- 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)
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
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
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
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
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
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
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.
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)
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.
- gcd(right, algorithm=None, **kwds)[source]#
Return the greatest common divisor of self and right.
INPUT:
right– polynomialalgorithm-'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
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
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
- 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: defaultRealFieldprecision).
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
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
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
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
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
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
- 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]
- hamming_weight()[source]#
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
>>> 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
The method
hamming_weight()is an alias:sage: f.hamming_weight() 101
>>> from sage.all import * >>> f.hamming_weight() 101
- in_subalgebra(J, algorithm='algebra_containment')[source]#
Return whether this polynomial is contained in the subalgebra generated by
JINPUT:
J– list of elements of the parent polynomial ringalgorithm– can be"algebra_containment"(the default),"inSubring", or"groebner"."algebra_containment": use Singular’salgebra_containmentfunction, 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’sinSubringfunction, 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
- 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
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
- 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
- is_constant()[source]#
Return
Trueif 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
- is_homogeneous()[source]#
Return
Trueif 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
- is_monomial()[source]#
Return
Trueif 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
- is_squarefree()[source]#
Return
Trueif 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
- is_term()[source]#
Return
Trueifselfis 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
- is_univariate()[source]#
Return
Trueifselfis a univariate polynomial, that is ifselfcontains 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
- is_zero()[source]#
Return
Trueif 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
- iterator_exp_coeff(as_ETuples=True)[source]#
Iterate over
selfas pairs of ((E)Tuple, coefficient).INPUT:
as_ETuples– (default:True) ifTrueiterate 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)]
- 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
- lcm(g)[source]#
Return the least common multiple of
selfand \(g\).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
- lift(I)[source]#
given an ideal
I = (f_1,...,f_r)and someg (== self)inI, finds_1,...,s_rsuch thatg = s_1 f_1 + ... + s_r f_r.A
ValueErrorexception is raised ifg (== self)does not belong toI.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
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
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]
- lm()[source]#
Returns the lead monomial of
selfwith respect to the term order ofself.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
- 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: defaultRealFieldprecision).
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
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
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
- 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– an integer.prec– desired floating point precision (default: defaultRealFieldprecision).
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
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
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
- 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
- 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
coefficientwhich 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
- 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]
- number_of_terms()[source]#
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
>>> 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
The method
hamming_weight()is an alias:sage: f.hamming_weight() 101
>>> from sage.all import * >>> f.hamming_weight() 101
- 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
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
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
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
- 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
- quo_rem(right)[source]#
Return quotient and remainder of
selfandright.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)
- 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
rsuch that:self–ris in the ideal generated byI.No term in
ris divisible by any of the leading monomials ofI.
The result
ris canonical if:Iis an ideal, and Sage can compute a Groebner basis of it.Iis a list/set/iterable that is a (strong) Groebner basis for the term order ofself. (A strong Groebner basis is such that for every leading termtof the ideal generated byI, there exists an elementgofIsuch that the leading term ofgdividest.)
The result
ris implementation-dependent (and possibly order-dependent) otherwise. IfIis an ideal and no Groebner basis can be computed, its list of generatorsI.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
\(\ZZ\) 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
The reduction is not canonical when
Iis 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
- 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– polynomialvariable– 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
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
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
- sub_m_mul_q(m, q)[source]#
Return
self - m*q, wheremmust be a monomial andqa polynomial.INPUT:
m– a monomialq– 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
- 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
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
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
- total_degree(std_grading=False)[source]#
Return the total degree of
self, which is the maximum degree of all monomials inself.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
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
- univariate_polynomial(R=None)[source]#
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
ValueErrorexception is raised. This is checked using theis_univariate()method. The new Polynomial is over the same base ring as the givenMPolynomialand in the variablexif no ringRis 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
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
- variable(i=0)[source]#
Return the \(i\)-th variable occurring in
self. The index \(i\) is the index inself.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
- 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)
- 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
- sage.rings.polynomial.multi_polynomial_libsingular.unpickle_MPolynomial_libsingular(R, d)[source]#
Deserialize an
MPolynomial_libsingularobjectINPUT:
R– the base ringd– a Python dictionary as returned byMPolynomial_libsingular.dict()
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