Polynomials ring¶

Multivariate Polynomials ring¶

Operations over polynomial rings.

class simula.polyring.polyring.PolynomialRing(domain, symbols=None, order=DegreeLexicographicOrder(), **kwargs)¶

Multivariate polynomial ring.

INPUT:

Parameters

domain – a domain (eg. QQ, RR, CC, ZZ, GF(p))
symbols – a sequence of symbols
order – (default ‘deglex’) a monomial ordering e.g. ‘lex’, ‘deglex’, ‘degrevlex’
kwargs –

EXAMPLES:

simula : R = PolynomialRing(QQ, "x, y, z", order="lex")
simula : x, y ,z = R.gens

These two lines are equivalent to the following code:

simula : R.<x, y ,z> = PolynomialRing(QQ, "x, y, z", order="lex")

By default the monomial ordering is "deglex", if don't need to change it, we can simplify again the notation.

simula : R.<x, y ,z> = QQ[]
simula : p1 = x^3*y-x*y^2-x-z; p1
x^3*y - x*y^2 - x - z
simula : p1.lcm(x-y-y*z)
x^4*y - x^3*y^2*z - x^3*y^2 - x^2*y^2 - x^2 + x*y^3*z + x*y^3 + x*y*z + x*y - x*z + y*z^2 + y*z
simula : R.make_monic(6x^4-3x-1)
x^4 - 1/2x - 1/6
simula : I = R.ideal([x*y^2-y-z, x^2*z-y*x]); I
ideal generated by [x*y^2 - y - z, x^2*z - x*y] of Polynomial ring in x, y, z over QQ with deglex order

add(pol1, pol2)¶: Returns pol1 + pol2 in self.

change_ring(domain=None, symbols=None, order=None)¶: Returns a new polynomial ring with the new given domain domain.

characteristic()¶: Returns the characteristic self.

cyclotomic_polynomial(n)¶

Returns the n-th cyclotomic polynomial.

EXAMPLES:

simula : R.<x> = GF(5)[]
simula : R.cyclotomic_polynomial(3)
x^2 + x + 1
simula : R.cyclotomic_polynomial(6)
x^2 + 4x + 1

div(pol1, pol2)¶: Returns the quotient and the remainder of the division of pol1 by pol2 in self.

factor(pol)¶: Returns the factorisation of the polynomial pol.

gcd(pol1, pol2)¶: Returns the gcd of pol1 and pol2.

gcdex(pol1, pol2)¶: Returns the extended gcd of pol1 and pol2.

ideal(F)¶: Returns the ideal in self generated by F.

is_exact()¶: Tests if self is an exact domain.

is_field()¶: Tests if self is a field.

is_irreducible(pol)¶: Tests if pol is an irreducible polynomial.

lcm(pol1, pol2)¶: Returns the lcm of pol1 and pol2.

make_monic(pol)¶: Makes monic the polynomial pol.

monic(pol)¶: Makes monic the polynomial pol.

mul(pol1, pol2)¶: Returns pol1 * pol2 in self.

objgen()¶

Returns self and its generators.

EXAMPLES:

simula : ring = PolynomialRing(QQ, "x, y, z", order="lex"); ring
simula : R, gens = ring.objgen()
simula : R
Multivariate Polynomial Ring in x, y, z over QQ with lex order
simula : gens
(x, y, z)

pow(pol, n)¶: Returns pol1^n in self.

primitive_polynomials(deg)¶

Returns the primitive polynomials of degree deg if self is a finite polynomial ring.

EXAMPLES:

simula : R.<x> = GF(5)[]
simula : R.primitive_polynomials(3)
{x^2 + x + 2, x^2 + 4x + 2, x^2 + 3x + 3, x^2 + 2x + 3}

quo(pol1, pol2)¶: Returns the quotient of the division of pol1 by pol2 in self.

random_irreducible(n)¶: Returns a random irreducible polynomial of degree n.

rem(pol1, pol2)¶: Returns the remainder of the division of pol1 by pol2 in self.

roots(f)¶: Returns the roots of the polynomial n.

sub(pol1, pol2)¶: Returns pol1 - pol2 in self.

univariate_ring(x)¶: Returns a univariate ring in x which has the same domain as self.

Groeber Bases¶

Operations over Groebner Bases.

class simula.polyring.groebner.Ideal(F, symbols=None, domain=None, order=None, *, ring=None)¶

Ideal generated by a set of polynomials F.

Parameters

F – a list of polynomials
symbols – (optional) list of variables
domain – (optional) a domain e.g. QQ, RR, ZZ
order – (optional) a monomial ordering e.g. ‘lex’, ‘deglex’, ‘degrevlex’
ring – (optional) a polynomial ring.

EXEMPLES:

simula : R.<x, y, z> = QQ[]
simula : R
Multivariate Polynomial Ring in x, y, z over QQ with deglex order
simula : I = ideal([x^2*y-z, x*y-1]); I
ideal generated by [x^2*y - z, x*y - 1] of Polynomial ring in x, y, z over QQ with deglex order
simula : J = (x^2*y-z, x*y-1) * R; J
ideal generated by [x^2*y
simula : I == J
True
simula : J.groebner_basis()
[y*z - 1, x - z]
simula : J.buchberger()
[x^2*y - z, x*y - 1, x - z, y*z - 1]
simula : J.homogenize('h')
ideal generated by [x^2*y - z*h^2, x*y - h^2] of Polynomial ring in x, y, z, h over QQ with deglex order
simula : J.reduce(x-y)
-y + z
simula : J.reduce(x^2*y-z + 2x*y-2)
0

basis()¶: Returns the basis of self.

basis_as_expr()¶: Returns the basis of self as an expression.

basis_is_groebner()¶: Tests if the given basis is a groebner basis of self.

buchberger()¶: Returns a groebner basis of self using a toy Buchberger algorithm.

change_ring(new_ring)¶: Returns a new ideal with the new polynomial ring.

groebner_basis()¶: Returns a reduced groebner basis of self.

groebner_basis_f5()¶: Returns a reduced groebner basis of self using the F5 algorithm.

homogenize(var=None)¶: Returns the ideal generated by the homogeneous polynomials of the basis of self.

is_homogeneous()¶: Tests if the polynomials in the basis of self are homogeneous.

is_in_radical_ideal(f)¶: Tests if f is in radical of self.

leading_ideal()¶: Returns the leading ideal of self.

normal_form(f, greobner=False)¶: Returns the normal form of f with respect to the basis of self.

reduce(f)¶: Reduces f with respect to the basis of self.

weak_normal_form(f, greobner=False)¶: Returns the weak normal form of f with respect to the basis of self.

simula.polyring.groebner.LC(f, symbols=None, **kwargs)¶: Returns the leading coefficient of f.

simula.polyring.groebner.LM(f, symbols=None, **kwargs)¶: Returns the leading monomial of f.

simula.polyring.groebner.LT(f, symbols=None, **kwargs)¶: Returns the leading term of f.

simula.polyring.groebner.groebner_basis(G, symbols=None, domain=None, order=None)¶: Returns a reduced groebner basis of the ideal generated by ``G`.

simula.polyring.groebner.groebner_f5(G, symbols=None, domain=None, order=None)¶: Returns a reduce groebner basis using the F5 algorithm.

simula.polyring.groebner.ideal¶: alias of simula.polyring.groebner.Ideal

simula.polyring.groebner.leading_coefficient(f, symbols=None, **kwargs)¶: Returns the leading coefficient of f.

simula.polyring.groebner.leading_ideal(I)¶

Returns the leading ideal of I.

Parameters: I (simula.polyring.groebner.Ideal) –

simula.polyring.groebner.leading_monom(f, symbols=None, **kwargs)¶: Returns the leading monomial of f.

simula.polyring.groebner.leading_term(f, symbols=None, **kwargs)¶: Returns the leading term of f.

simula.polyring.groebner.normal_form(f, G, symbols=None, domain=Rational Numbers, order=DegreeLexicographicOrder())¶: Returns the normal form of f in G.

simula.polyring.groebner.spoly(f, g, symbols=None, domain=Rational Numbers, order=DegreeLexicographicOrder())¶: Returns the S-polynomial of f and g .

simula.polyring.groebner.weak_normal_form(f, G, symbols=None, domain=Rational Numbers, order=DegreeLexicographicOrder())¶: Returns the weak normal form of f in G