Educational versions of Groebner basis algorithms: triangular factorization#
In this file is the implementation of two algorithms in [Laz1992].
The main algorithm is Triangular
; a secondary algorithm, necessary for the
first, is ElimPolMin
. As per Lazard’s formulation, the implementation works
with any term ordering, not only lexicographic.
Lazard does not specify a few of the subalgorithms implemented as the functions
is_triangular
,is_linearly_dependent
, andlinear_representation
.
The implementations are not hard, and the choice of algorithm is described with the relevant function.
No attempt was made to optimize these algorithms as the emphasis of this implementation is a clean and easy presentation.
Examples appear with the appropriate function.
AUTHORS:
John Perry (2009-02-24): initial version, but some words of documentation were stolen shamelessly from Martin Albrecht’s
toy_buchberger.py
.
- sage.rings.polynomial.toy_variety.coefficient_matrix(polys)#
Generate the matrix
M
whose entries are the coefficients ofpolys
.The entries of row
i
ofM
consist of the coefficients ofpolys[i]
.INPUT:
polys
– a list/tuple of polynomials
OUTPUT:
A matrix
M
of the coefficients ofpolys
EXAMPLES:
sage: from sage.rings.polynomial.toy_variety import coefficient_matrix sage: R.<x,y> = PolynomialRing(QQ) sage: coefficient_matrix([x^2 + 1, y^2 + 1, x*y + 1]) # needs sage.modules [1 0 0 1] [0 0 1 1] [0 1 0 1]
Note
This function may be merged with
sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic.coefficient_matrix()
in the future.
- sage.rings.polynomial.toy_variety.elim_pol(B, n=-1)#
Find the unique monic polynomial of lowest degree and lowest variable in the ideal described by
B
.For the purposes of the triangularization algorithm, it is necessary to preserve the ring, so
n
specifies which variable to check. By default, we check the last one, which should also be the smallest.The algorithm may not work if you are trying to cheat:
B
should describe the Groebner basis of a zero-dimensional ideal. However, it is not necessary for the Groebner basis to be lexicographic.The algorithm is taken from a 1993 paper by Lazard [Laz1992].
INPUT:
B
– a list/tuple of polynomials or a multivariate polynomial idealn
– the variable to check (see above) (default:-1
)
EXAMPLES:
sage: # needs sage.rings.finite_rings sage: from sage.misc.verbose import set_verbose sage: set_verbose(0) sage: from sage.rings.polynomial.toy_variety import elim_pol sage: R.<x,y,z> = PolynomialRing(GF(32003)) sage: p1 = x^2*(x-1)^3*y^2*(z-3)^3 sage: p2 = z^2 - z sage: p3 = (x-2)^2*(y-1)^3 sage: I = R.ideal(p1,p2,p3) sage: elim_pol(I.groebner_basis()) # needs sage.libs.singular z^2 - z
- sage.rings.polynomial.toy_variety.is_linearly_dependent(polys)#
Decide whether the polynomials of
polys
are linearly dependent.Here
polys
is a collection of polynomials.The algorithm creates a matrix of coefficients of the monomials of
polys
. It computes the echelon form of the matrix, then checks whether any of the rows is the zero vector.Essentially this relies on the fact that the monomials are linearly independent, and therefore is building a linear map from the vector space of the monomials to the canonical basis of
R^n
, wheren
is the number of distinct monomials inpolys
. There is a zero vector iff there is a linear dependence amongpolys
.The case where
polys=[]
is considered to be not linearly dependent.INPUT:
polys
– a list/tuple of polynomials
OUTPUT:
True
if the elements ofpolys
are linearly dependent;False
otherwise.EXAMPLES:
sage: from sage.rings.polynomial.toy_variety import is_linearly_dependent sage: R.<x,y> = PolynomialRing(QQ) sage: B = [x^2 + 1, y^2 + 1, x*y + 1] sage: p = 3*B[0] - 2*B[1] + B[2] sage: is_linearly_dependent(B + [p]) # needs sage.modules True sage: p = x*B[0] sage: is_linearly_dependent(B + [p]) # needs sage.modules False sage: is_linearly_dependent([]) False sage: R.<x> = PolynomialRing(QQ) sage: B = [x^147 + x^99, ....: 2*x^123 + x^75, ....: x^147 + 2*x^123 + 2*x^75, ....: 2*x^147 + x^99 + x^75] sage: is_linearly_dependent(B) True
- sage.rings.polynomial.toy_variety.is_triangular(B)#
Check whether the basis
B
of an ideal is triangular.That is: check whether the largest variable in
B[i]
with respect to the ordering of the base ringR
isR.gens()[i]
.The algorithm is based on the definition of a triangular basis, given by Lazard in 1992 in [Laz1992].
INPUT:
B
– a list/tuple of polynomials or a multivariate polynomial ideal
OUTPUT:
True
if the basis is triangular;False
otherwise.EXAMPLES:
sage: from sage.rings.polynomial.toy_variety import is_triangular sage: R.<x,y,z> = PolynomialRing(QQ) sage: p1 = x^2*y + z^2 sage: p2 = y*z + z^3 sage: p3 = y+z sage: is_triangular(R.ideal(p1,p2,p3)) False sage: p3 = z^2 - 3 sage: is_triangular(R.ideal(p1,p2,p3)) True
- sage.rings.polynomial.toy_variety.linear_representation(p, polys)#
Assuming that
p
is a linear combination ofpolys
, determine coefficients that describe the linear combination.This probably does not work for any inputs except
p
, a polynomial, andpolys
, a sequence of polynomials. Ifp
is not in fact a linear combination ofpolys
, the function raises an exception.The algorithm creates a matrix of coefficients of the monomials of
polys
andp
, with the coefficients ofp
in the last row. It augments this matrix with the appropriate identity matrix, then computes the echelon form of the augmented matrix. The last row should contain zeroes in the first columns, and the last columns contain a linear dependence relation. Solving for the desired linear relation is straightforward.INPUT:
p
– a polynomialpolys
– a list/tuple of polynomials
OUTPUT:
If
n == len(polys)
, returns[a[0],a[1],...,a[n-1]]
such thatp == a[0]*poly[0] + ... + a[n-1]*poly[n-1]
.EXAMPLES:
sage: # needs sage.modules sage.rings.finite_rings sage: from sage.rings.polynomial.toy_variety import linear_representation sage: R.<x,y> = PolynomialRing(GF(32003)) sage: B = [x^2 + 1, y^2 + 1, x*y + 1] sage: p = 3*B[0] - 2*B[1] + B[2] sage: linear_representation(p, B) [3, 32001, 1]
- sage.rings.polynomial.toy_variety.triangular_factorization(B, n=-1)#
Compute the triangular factorization of the Groebner basis
B
of an ideal.This will not work properly if
B
is not a Groebner basis!The algorithm used is that described in a 1992 paper by Daniel Lazard [Laz1992]. It is not necessary for the term ordering to be lexicographic.
INPUT:
B
– a list/tuple of polynomials or a multivariate polynomial idealn
– the recursion parameter (default:-1
)
OUTPUT:
A list
T
of triangular setsT_0
,T_1
, etc.EXAMPLES:
sage: # needs sage.rings.finite_rings sage: from sage.misc.verbose import set_verbose sage: set_verbose(0) sage: from sage.rings.polynomial.toy_variety import triangular_factorization sage: R.<x,y,z> = PolynomialRing(GF(32003)) sage: p1 = x^2*(x-1)^3*y^2*(z-3)^3 sage: p2 = z^2 - z sage: p3 = (x-2)^2*(y-1)^3 sage: I = R.ideal(p1,p2,p3) sage: triangular_factorization(I.groebner_basis()) # needs sage.libs.singular [[x^2 - 4*x + 4, y, z], [x^5 - 3*x^4 + 3*x^3 - x^2, y - 1, z], [x^2 - 4*x + 4, y, z - 1], [x^5 - 3*x^4 + 3*x^3 - x^2, y - 1, z - 1]]