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
This is an implementation of the Macaulay Resultant. It computes the resultant of universal polynomials as well as polynomials with constant coefficients. This is a project done in sage days 55. It’s based on the implementation in Maple by Manfred Minimair, which in turn is based on the references listed below: It calculates the Macaulay resultant for a list of polynomials, up to sign!
REFERENCES:
[CLO] | D. Cox, J. Little, D. O’Shea. Using Algebraic Geometry. Springer, 2005. |
[Can] | J. Canny. Generalised characteristic polynomials. J. Symbolic Comput. Vol. 9, No. 3, 1990, 241–250. |
[Mac] | F.S. Macaulay. The algebraic theory of modular systems Cambridge university press, 1916. |
AUTHORS:
INPUT:
works when args[0] is the list of polynomials, or args is itself the list of polynomials
kwds:
if True function creates sparse matrices.
OUTPUT:
Todo
Working with sparse matrices should usually give faster results, but with the current implementation it actually works slower. There should be a way to improve performance with regards to this.
EXAMPLES:
The number of polynomials has to match the number of variables:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant([y,x+z])
Traceback (most recent call last):
...
TypeError: number of polynomials(= 2) must equal number of variables (= 3)
The polynomials need to be all homogeneous:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant([y, x+z, z+x^3])
Traceback (most recent call last):
...
TypeError: resultant for non-homogeneous polynomials is not supported
All polynomials must be in the same ring:
sage: S.<x,y> = PolynomialRing(QQ, 2)
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: S.macaulay_resultant([y, z+x])
Traceback (most recent call last):
...
TypeError: not all inputs are polynomials in the calling ring
The following example recreates Proposition 2.10 in Ch.3 in [CLO]:
sage: K.<x,y> = PolynomialRing(ZZ, 2)
sage: flist,R = K._macaulay_resultant_universal_polynomials([1,1,2])
sage: R.macaulay_resultant(flist)
u2^2*u4^2*u6 - 2*u1*u2*u4*u5*u6 + u1^2*u5^2*u6 - u2^2*u3*u4*u7 + u1*u2*u3*u5*u7 + u0*u2*u4*u5*u7 - u0*u1*u5^2*u7 + u1*u2*u3*u4*u8 - u0*u2*u4^2*u8 - u1^2*u3*u5*u8 + u0*u1*u4*u5*u8 + u2^2*u3^2*u9 - 2*u0*u2*u3*u5*u9 + u0^2*u5^2*u9 - u1*u2*u3^2*u10 + u0*u2*u3*u4*u10 + u0*u1*u3*u5*u10 - u0^2*u4*u5*u10 + u1^2*u3^2*u11 - 2*u0*u1*u3*u4*u11 + u0^2*u4^2*u11
The following example degenerates into the determinant of a matrix:
sage: K.<x,y> = PolynomialRing(ZZ, 2)
sage: flist,R = K._macaulay_resultant_universal_polynomials([1,1,1])
sage: R.macaulay_resultant(flist)
-u2*u4*u6 + u1*u5*u6 + u2*u3*u7 - u0*u5*u7 - u1*u3*u8 + u0*u4*u8
The following example is by Patrick Ingram(arxiv:1310.4114):
sage: U = PolynomialRing(ZZ,'y',2); y0,y1 = U.gens()
sage: R = PolynomialRing(U,'x',3); x0,x1,x2 = R.gens()
sage: f0 = y0*x2^2 - x0^2 + 2*x1*x2
sage: f1 = y1*x2^2 - x1^2 + 2*x0*x2
sage: f2 = x0*x1 - x2^2
sage: flist = [f0,f1,f2]
sage: R.macaulay_resultant([f0,f1,f2])
y0^2*y1^2 - 4*y0^3 - 4*y1^3 + 18*y0*y1 - 27
a simple example with constant rational coefficients:
sage: R.<x,y,z,w> = PolynomialRing(QQ,4)
sage: R.macaulay_resultant([w,z,y,x])
1
an example where the resultant vanishes:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant([x+y,y^2,x])
0
an example of bad reduction at a prime :
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant([y,x^3+25*y^2*x,5*z])
125
The input can given as an unpacked list of polynomials:
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant(y,x^3+25*y^2*x,5*z)
125
an example when the coefficients live in a finite field:
sage: F = FiniteField(11)
sage: R.<x,y,z,w> = PolynomialRing(F,4)
sage: R.macaulay_resultant([z,x^3,5*y,w])
4
example when the denominator in the algorithm vanishes(in this case the resultant is the constant term of the quotient of char polynomials of numerator/denominator):
sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: R.macaulay_resultant([y, x+z, z^2])
-1
when there are only 2 polynomials, macaulay resultant degenerates to the traditional resultant:
sage: R.<x> = PolynomialRing(QQ,1)
sage: f = x^2+1; g = x^5+1
sage: fh = f.homogenize()
sage: gh = g.homogenize()
sage: RH = fh.parent()
sage: f.resultant(g) == RH.macaulay_resultant([fh,gh])
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 - x - 1/14*z - 1
Stacked rings:
sage: R = QQ['x,y']
sage: S = R['t,u']
sage: S.random_element(degree=2, terms=1)
-1/2*x^2 - 1/4*x*y - 3*y^2 + 4*y
sage: S.random_element(degree=2, terms=1)
(-x^2 - 2*y^2 - 1/3*x + 2*y + 9)*u^2
Default values apply if no degree and/or number of terms is provided:
sage: random_matrix(QQ['x,y,z'], 2, 2)
[357*x^2 + 1/4*y^2 + 2*y*z + 2*z^2 + 28*x 2*x*y + 3/2*y^2 + 2*y*z - 2*z^2 - z]
[ x*y - y*z + 2*z^2 -x^2 - 4/3*x*z + 2*z^2 - x + 4*y]
sage: random_matrix(QQ['x,y,z'], 2, 2, terms=1, degree=2)
[ 1/2*y -1/4*x]
[ 1/2 1/3*x]
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^2 + x
To produce a dense polynomial, pick terms=Infinity:
sage: P.<x,y,z> = GF(127)[]
sage: P.random_element(degree=2, terms=Infinity)
-55*x^2 - 51*x*y + 5*y^2 + 55*x*z - 59*y*z + 20*z^2 + 19*x - 55*y - 28*z + 17
sage: P.random_element(degree=3, terms=Infinity)
-54*x^3 + 15*x^2*y - x*y^2 - 15*y^3 + 61*x^2*z - 12*x*y*z + 20*y^2*z - 61*x*z^2 - 5*y*z^2 + 62*z^3 + 15*x^2 - 47*x*y + 31*y^2 - 14*x*z + 29*y*z + 13*z^2 + 61*x - 40*y - 49*z + 30
sage: P.random_element(degree=3, terms=Infinity, choose_degree=True)
57*x^3 - 58*x^2*y + 21*x*y^2 + 36*y^3 + 7*x^2*z - 57*x*y*z + 8*y^2*z - 11*x*z^2 + 7*y*z^2 + 6*z^3 - 38*x^2 - 18*x*y - 52*y^2 + 27*x*z + 4*y*z - 51*z^2 - 63*x + 7*y + 48*z + 14
The number of terms is silently reduced to the maximum available if more terms are requested:
sage: P.<x,y,z> = GF(127)[]
sage: P.random_element(degree=2, terms=1000)
5*x^2 - 10*x*y + 10*y^2 - 44*x*z + 31*y*z + 19*z^2 - 42*x - 50*y - 49*z - 60
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')
Return the Weyl algebra generated from self.
EXAMPLES:
sage: R = QQ['x,y,z']
sage: W = R.weyl_algebra(); W
Differential Weyl algebra of polynomials in x, y, z over Rational Field
sage: W.polynomial_ring() == R
True