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, and
linear_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)[source]¶

Generate the matrix M whose entries are the coefficients of polys.

The entries of row i of M consist of the coefficients of polys[i].

INPUT:

polys – list/tuple of polynomials

OUTPUT: a matrix M of the coefficients of polys

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> from sage.rings.polynomial.toy_variety import coefficient_matrix
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> coefficient_matrix([x**Integer(2) + Integer(1), y**Integer(2) + Integer(1), x*y + Integer(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)[source]¶

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 – list/tuple of polynomials or a multivariate polynomial ideal
n – the variable to check (see above) (default: -1)

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> from sage.misc.verbose import set_verbose
>>> set_verbose(Integer(0))
>>> from sage.rings.polynomial.toy_variety import elim_pol
>>> R = PolynomialRing(GF(Integer(32003)), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p1 = x**Integer(2)*(x-Integer(1))**Integer(3)*y**Integer(2)*(z-Integer(3))**Integer(3)
>>> p2 = z**Integer(2) - z
>>> p3 = (x-Integer(2))**Integer(2)*(y-Integer(1))**Integer(3)
>>> I = R.ideal(p1,p2,p3)
>>> elim_pol(I.groebner_basis())                                              # needs sage.libs.singular
z^2 - z

sage.rings.polynomial.toy_variety.is_linearly_dependent(polys)[source]¶

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, where n is the number of distinct monomials in polys. There is a zero vector iff there is a linear dependence among polys.

The case where polys=[] is considered to be not linearly dependent.

INPUT:

polys – list/tuple of polynomials

OUTPUT:

True if the elements of polys are linearly dependent; False otherwise.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.polynomial.toy_variety import is_linearly_dependent
>>> R = PolynomialRing(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> B = [x**Integer(2) + Integer(1), y**Integer(2) + Integer(1), x*y + Integer(1)]
>>> p = Integer(3)*B[Integer(0)] - Integer(2)*B[Integer(1)] + B[Integer(2)]
>>> is_linearly_dependent(B + [p])                                            # needs sage.modules
True
>>> p = x*B[Integer(0)]
>>> is_linearly_dependent(B + [p])                                            # needs sage.modules
False
>>> is_linearly_dependent([])
False
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> B = [x**Integer(147) + x**Integer(99),
...      Integer(2)*x**Integer(123) + x**Integer(75),
...      x**Integer(147) + Integer(2)*x**Integer(123) + Integer(2)*x**Integer(75),
...      Integer(2)*x**Integer(147) + x**Integer(99) + x**Integer(75)]
>>> is_linearly_dependent(B)
True

sage.rings.polynomial.toy_variety.is_triangular(B)[source]¶

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 ring R is R.gens()[i].

The algorithm is based on the definition of a triangular basis, given by Lazard in 1992 in [Laz1992].

INPUT:

B – list/tuple of polynomials or a multivariate polynomial ideal

OUTPUT: True if the basis is triangular; False otherwise

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.rings.polynomial.toy_variety import is_triangular
>>> R = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p1 = x**Integer(2)*y + z**Integer(2)
>>> p2 = y*z + z**Integer(3)
>>> p3 = y+z
>>> is_triangular(R.ideal(p1,p2,p3))
False
>>> p3 = z**Integer(2) - Integer(3)
>>> is_triangular(R.ideal(p1,p2,p3))
True

sage.rings.polynomial.toy_variety.linear_representation(p, polys)[source]¶

Assuming that p is a linear combination of polys, determine coefficients that describe the linear combination.

This probably does not work for any inputs except p, a polynomial, and polys, a sequence of polynomials. If p is not in fact a linear combination of polys, the function raises an exception.

The algorithm creates a matrix of coefficients of the monomials of polys and p, with the coefficients of p 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 polynomial
polys – list/tuple of polynomials

OUTPUT:

If n == len(polys), returns [a[0],a[1],...,a[n-1]] such that p == a[0]*poly[0] + ... + a[n-1]*poly[n-1].

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> # needs sage.modules sage.rings.finite_rings
>>> from sage.rings.polynomial.toy_variety import linear_representation
>>> R = PolynomialRing(GF(Integer(32003)), names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> B = [x**Integer(2) + Integer(1), y**Integer(2) + Integer(1), x*y + Integer(1)]
>>> p = Integer(3)*B[Integer(0)] - Integer(2)*B[Integer(1)] + B[Integer(2)]
>>> linear_representation(p, B)
[3, 32001, 1]

sage.rings.polynomial.toy_variety.triangular_factorization(B, n=-1)[source]¶

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 – list/tuple of polynomials or a multivariate polynomial ideal
n – the recursion parameter (default: -1)

OUTPUT: list T of triangular sets T_0, T_1, etc.

EXAMPLES:

Sage

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]]

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> from sage.misc.verbose import set_verbose
>>> set_verbose(Integer(0))
>>> from sage.rings.polynomial.toy_variety import triangular_factorization
>>> R = PolynomialRing(GF(Integer(32003)), names=('x', 'y', 'z',)); (x, y, z,) = R._first_ngens(3)
>>> p1 = x**Integer(2)*(x-Integer(1))**Integer(3)*y**Integer(2)*(z-Integer(3))**Integer(3)
>>> p2 = z**Integer(2) - z
>>> p3 = (x-Integer(2))**Integer(2)*(y-Integer(1))**Integer(3)
>>> I = R.ideal(p1,p2,p3)
>>> 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]]