Sage implements multivariate polynomial rings through several backends. This generic implementation uses the classes PolyDict and ETuple to construct a dictionary with exponent tuples as keys and coefficients as values.
AUTHORS:
EXAMPLES:
We construct the Frobenius morphism on over
:
sage: R, (x,y,z) = PolynomialRing(GF(5), 3, 'xyz').objgens()
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
Multivariable polynomial ring.
EXAMPLES:
sage: R = PolynomialRing(Integers(12), 'x', 5); R
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Ring of integers modulo 12
sage: loads(R.dumps()) == R
True
Convert x to an element of this multivariate polynomial ring, possibly non-canonically.
EXAMPLES:
We create a Macaulay2 multivariate polynomial via ideal arithmetic, then convert it into R.
sage: R.<x,y> = PolynomialRing(QQ, 2) # optional
sage: I = R.ideal([x^3 + y, y]) # optional
sage: S = I._macaulay2_() # optional
sage: T = S*S*S # optional
sage: U = T.gens().entries().flatten() # optional
sage: f = U[2]; f # optional
x^6*y+2*x^3*y^2+y^3
sage: R(repr(f)) # optional
x^6*y + 2*x^3*y^2 + y^3
Some other subtle conversions. We create polynomial rings in 2 variables over the rationals, integers, and a finite field.
sage: R.<x,y> = QQ[]
sage: S.<x,y> = ZZ[]
sage: T.<x,y> = GF(7)[]
We convert from integer polynomials to rational polynomials, and back:
sage: f = R(S.0^2 - 4*S.1^3); f
-4*y^3 + x^2
sage: parent(f)
Multivariate Polynomial Ring in x, y over Rational Field
sage: parent(S(f))
Multivariate Polynomial Ring in x, y over Integer Ring
We convert from polynomials over the finite field.
sage: f = R(T.0^2 - 4*T.1^3); f
3*y^3 + x^2
sage: parent(f)
Multivariate Polynomial Ring in x, y over Rational Field
We dump and load the polynomial ring S:
sage: S2 = loads(dumps(S))
sage: S2 == S
True
Coerce works and gets the right parent.
sage: parent(S2._coerce_(S.0)) is S2
True
Conversion to reduce modulo a prime between rings with different variable names:
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = PolynomialRing(GF(7),2)
sage: f = x^2 + 2/3*y^3
sage: S(f)
3*b^3 + a^2
Conversion from symbolic variables:
sage: x,y,z = var('x,y,z')
sage: R = QQ[x,y,z]
sage: type(x)
<type 'sage.symbolic.expression.Expression'>
sage: type(R(x))
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f = R(x^3 + y^3 - z^3); f
x^3 + y^3 - z^3
sage: type(f)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: parent(f)
Multivariate Polynomial Ring in x, y, z over Rational Field
A more complicated symbolic and computational mix. Behind the scenes Singular and Maxima are doing the real work.
sage: R = QQ[x,y,z]
sage: f = (x^3 + y^3 - z^3)^10; f
(x^3 + y^3 - z^3)^10
sage: g = R(f); parent(g)
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: (f - g).expand()
0
It intelligently handles conversions from polynomial rings in a subset of the variables too.
sage: R = GF(5)['x,y,z']
sage: S = ZZ['y']
sage: R(7*S.0)
2*y
sage: T = ZZ['x,z']
sage: R(2*T.0 + 6*T.1 + T.0*T.1^2)
x*z^2 + 2*x + z
sage: R = QQ['t,x,y,z']
sage: S.<x> = ZZ['x']
sage: T.<z> = S['z']
sage: T
Univariate Polynomial Ring in z over Univariate Polynomial Ring in x over Integer Ring
sage: f = (x+3*z+5)^2; f
9*z^2 + (6*x + 30)*z + x^2 + 10*x + 25
sage: R(f)
x^2 + 6*x*z + 9*z^2 + 10*x + 30*z + 25
Arithmetic with a constant from a base ring:
sage: R.<u,v> = QQ[]
sage: S.<x,y> = R[]
sage: u^3*x^2 + v*y
u^3*x^2 + v*y
Stacked polynomial rings convert into constants if possible. First, the univariate case:
sage: R.<x> = QQ[]
sage: S.<u,v> = R[]
sage: S(u + 2)
u + 2
sage: S(u + 2).degree()
1
sage: S(x + 3)
x + 3
sage: S(x + 3).degree()
0
Second, the multivariate case:
sage: R.<x,y> = QQ[]
sage: S.<u,v> = R[]
sage: S(x + 2*y)
x + 2*y
sage: S(u + 2*v)
u + 2*v
Conversion from strings:
sage: R.<x,y> = QQ[]
sage: R('x+(1/2)*y^2')
1/2*y^2 + x
sage: S.<u,v> = ZZ[]
sage: S('u^2 + u*v + v^2')
u^2 + u*v + v^2
Foreign polynomial rings convert into the highest ring; the point here is that an element of T could convert to an element of R or an element of S; it is anticipated that an element of T is more likely to be “the right thing” and is historically consistent.
sage: R.<x,y> = QQ[]
sage: S.<u,v> = R[]
sage: T.<a,b> = QQ[]
sage: S(a + b)
u + v
Return a list of all monomials that divide t, coefficients are ignored.
INPUT:
OUTPUT: a list of monomials
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
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
Return False if a does not divide b and True otherwise.
INPUT:
EXAMPLES:
sage: P.<x,y,z>=MPolynomialRing(ZZ,3, order='degrevlex')
doctest:1: DeprecationWarning: MPolynomialRing is deprecated, use PolynomialRing instead!
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
TESTS:
sage: P.<x,y,z>=MPolynomialRing(ZZ,3, order='degrevlex')
sage: P.monomial_divides(P(1), P(0))
True
sage: P.monomial_divides(P(1), x)
True
LCM for monomials. Coefficients are ignored.
INPUT:
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_lcm(3/2*x*y,x)
x*y
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: R.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_lcm(x*y,R.gen())
x*y
sage: P.monomial_lcm(P(3/2),P(2/3))
1
sage: P.monomial_lcm(x,P(1))
x
Return True if h and g are pairwise prime. Both are treated as monomials.
INPUT:
EXAMPLES:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
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
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: Q.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_pairwise_prime(x^2*z^3, Q('y^4'))
True
sage: P.monomial_pairwise_prime(1/2*x^3*y^2, Q(0))
True
sage: P.monomial_pairwise_prime(P(1/2),x)
False
Return f/g, where both f and g are treated as monomials. Coefficients are ignored by default.
INPUT:
EXAMPLE:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ, 3, order='degrevlex')
sage: P.monomial_quotient(3/2*x*y,x)
y
sage: P.monomial_quotient(3/2*x*y,2*x,coeff=True)
3/4*y
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: R.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: P.monomial_quotient(x*y,x)
y
sage: P.monomial_quotient(x*y,R.gen())
y
sage: P.monomial_quotient(P(0),P(1))
0
sage: P.monomial_quotient(P(1),P(0))
...
ZeroDivisionError
sage: P.monomial_quotient(P(3/2),P(2/3), coeff=True)
9/4
sage: P.monomial_quotient(x,y) # Note the wrong result
x*y^-1
sage: P.monomial_quotient(x,P(1))
x
Note
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.
Try to find a g in G where g.lm() divides f. If found (g,flt) is returned, (0,0) otherwise, where flt is f/g.lm().
It is assumed that G is iterable and contains ONLY elements in self.
INPUT:
EXAMPLES:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, P(1/2) ]
sage: P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)
TESTS:
sage: from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict_domain
sage: P.<x,y,z>=MPolynomialRing_polydict_domain(QQ,3, order='degrevlex')
sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, P(1/2) ]
sage: P.monomial_reduce(P(0),G)
(0, 0)
sage: P.monomial_reduce(f,[P(0)])
(0, 0)