Base class for multivariate polynomial rings
Bases: sage.rings.ring.CommutativeRing
Create a polynomial ring in several variables over a commutative ring.
EXAMPLES:
sage: R.<x,y> = ZZ['x,y']; R
Multivariate Polynomial Ring in x, y over Integer Ring
sage: class CR(CommutativeRing):
... def __init__(self):
... CommutativeRing.__init__(self,self)
... def __call__(self,x):
... return None
sage: cr = CR()
sage: cr.is_commutative()
True
sage: cr['x,y']
Multivariate Polynomial Ring in x, y over <class '....CR_with_category'>
TESTS:
Check that containment works correctly (ticket #10355):
sage: A1.<a> = PolynomialRing(QQ)
sage: A2.<a,b> = PolynomialRing(QQ)
sage: 3 in A2
True
sage: A1(a) in A2
True
Return a new multivariate polynomial ring which isomorphic to self, but has a different ordering given by the parameter ‘order’ or names given by the parameter ‘names’.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(GF(127),3,order='lex')
sage: x > y^2
True
sage: Q.<x,y,z> = P.change_ring(order='degrevlex')
sage: x > y^2
False
Return the characteristic of this polynomial ring.
EXAMPLES:
sage: R = PolynomialRing(QQ, 'x', 3)
sage: R.characteristic()
0
sage: R = PolynomialRing(GF(7),'x', 20)
sage: R.characteristic()
7
Return the completion of self with respect to the ideal generated by the variable(s) p.
INPUT:
EXAMPLES:
sage: P.<x,y,z,w> = PolynomialRing(ZZ)
sage: P.completion((w,x,y))
Multivariate Power Series Ring in w, x, y over Univariate Polynomial Ring in z over Integer Ring
sage: P.completion((w,x,y,z))
Multivariate Power Series Ring in w, x, y, z over Integer Ring
sage: H = PolynomialRing(PolynomialRing(ZZ,3,'z'),4,'f'); H
Multivariate Polynomial Ring in f0, f1, f2, f3 over
Multivariate Polynomial Ring in z0, z1, z2 over Integer Ring
sage: H.completion(H.gens())
Multivariate Power Series Ring in f0, f1, f2, f3 over
Multivariate Polynomial Ring in z0, z1, z2 over Integer Ring
sage: H.completion(H.gens()[2])
Power Series Ring in f2 over
Multivariate Polynomial Ring in f0, f1, f3 over
Multivariate Polynomial Ring in z0, z1, z2 over Integer Ring
Returns a functor F and base ring R such that F(R) == self.
EXAMPLES:
sage: S = ZZ['x,y']
sage: F, R = S.construction(); R
Integer Ring
sage: F
MPoly[x,y]
sage: F(R) == S
True
sage: F(R) == ZZ['x']['y']
False
Return the irrelevant ideal of this multivariate polynomial ring, which is the ideal generated by all of the indeterminate generators of this ring.
EXAMPLES:
sage: R.<x,y,z> = QQ[]
sage: R.irrelevant_ideal()
Ideal (x, y, z) of Multivariate Polynomial Ring in x, y, z over Rational Field
Return True if this multivariate polynomial ring is a field, i.e., it is a ring in 0 generators over a field.
EXAMPLES:
sage: ZZ['x,y'].is_integral_domain()
True
sage: Integers(8)['x,y'].is_integral_domain()
False
EXAMPLES:
sage: ZZ['x,y'].is_noetherian()
True
sage: Integers(8)['x,y'].is_noetherian()
True
Return a random polynomial of at most degree and at most
terms.
First monomials are chosen uniformly random from the set of all
possible monomials of degree up to (inclusive). This means
that it is more likely that a monomial of degree
appears than
a monomial of degree
because the former class is bigger.
Exactly distinct monomials are chosen this way and each one gets
a random coefficient (possibly zero) from the base ring assigned.
The returned polynomial is the sum of this list of terms.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: P.random_element(2, 5)
-6/5*x^2 + 2/3*z^2 - 1
sage: P.random_element(2, 5, choose_degree=True)
-1/4*x*y - 1/5*x*z - 1/14*y*z - z^2
Stacked rings:
sage: R = QQ['x,y']
sage: S = R['t,u']
sage: S.random_element(degree=2, terms=1)
-3*x*y + 5/2*y^2 - 1/2*x - 1/4*y + 4
sage: S.random_element(degree=2, terms=1)
(-1/2*x^2 - x*y - 2/7*y^2 + 3/2*x - y)*t*u
Default values apply if no degree and/or number of terms is provided:
sage: random_matrix(QQ['x,y,z'], 2, 2)
[ 2*y^2 - 2/27*y*z - z^2 + 2*z 1/2*x*y - 1/2*y^2 + 2*x - 2*y]
[-1/27*x^2 + 2/5*y^2 - 1/10*z^2 - 2*z -13*y^2 + 2/3*z^2 + 2*y]
sage: random_matrix(QQ['x,y,z'], 2, 2, terms=1, degree=2)
[-1/4*x 1/2]
[ 1/3*x x*y]
sage: P.random_element(0, 1)
-1
sage: P.random_element(2, 0)
0
sage: R.<x> = PolynomialRing(Integers(3), 1)
sage: R.random_element()
x + 1
Remove a variable or sequence of variables from self.
If order is not specified, then the subring inherits the term order of the original ring, if possible.
EXAMPLES:
sage: P.<x,y,z,w> = PolynomialRing(ZZ)
sage: P.remove_var(z)
Multivariate Polynomial Ring in x, y, w over Integer Ring
sage: P.remove_var(z,x)
Multivariate Polynomial Ring in y, w over Integer Ring
sage: P.remove_var(y,z,x)
Univariate Polynomial Ring in w over Integer Ring
Removing all variables results in the base ring:
sage: P.remove_var(y,z,x,w)
Integer Ring
If possible, the term order is kept:
sage: R.<x,y,z,w> = PolynomialRing(ZZ, order=’deglex’) sage: R.remove_var(y).term_order() Degree lexicographic term order
sage: R.<x,y,z,w> = PolynomialRing(ZZ, order=’lex’) sage: R.remove_var(y).term_order() Lexicographic term order
Be careful with block orders when removing variables:
sage: R.<x,y,z,u,v> = PolynomialRing(ZZ, order='deglex(2),lex(3)')
sage: R.remove_var(x,y,z)
Traceback (most recent call last):
...
ValueError: impossible to use the original term order (most likely because it was a block order). Please specify the term order for the subring
sage: R.remove_var(x,y,z, order='degrevlex')
Multivariate Polynomial Ring in u, v over Integer Ring
Return structured string representation of self.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ,order=TermOrder('degrevlex',1)+TermOrder('lex',2))
sage: print P.repr_long()
Polynomial Ring
Base Ring : Rational Field
Size : 3 Variables
Block 0 : Ordering : degrevlex
Names : x
Block 1 : Ordering : lex
Names : y, z
Return a univariate polynomial ring whose base ring comprises all but one variables of self.
INPUT:
EXAMPLE:
sage: P.<x,y,z> = QQ[]
sage: P.univariate_ring(y)
Univariate Polynomial Ring in y over Multivariate Polynomial Ring in x, z over Rational Field
Returns the list of variable names of this and its base rings, as if it were a single multi-variate polynomial.
EXAMPLES:
sage: R = QQ['x,y']['z,w']
sage: R.variable_names_recursive()
('x', 'y', 'z', 'w')
sage: R.variable_names_recursive(3)
('y', 'z', 'w')