In this file is the implementation of two algorithms in [Laz92].
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
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:
REFERENCES:
| [Laz92] | (1, 2, 3, 4) Daniel Lazard, Solving Zero-dimensional Algebraic Systems, in Journal of Symbolic Computation (1992) vol. 13, pp. 117-131 |
Generates 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:
OUTPUT:
A matrix M of the coefficients of polys.
EXAMPLE:
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])
[1 0 0 1]
[0 0 1 1]
[0 1 0 1]
Note
This function may be merged with sage.crypto.mq.MPolynomialSystem_generic.coefficient_matrix() in the future.
Finds 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 [Laz92].
INPUT:
EXAMPLE:
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())
z^2 - z
Decides 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:
OUTPUT:
True if the elements of polys are linearly dependent; False otherwise.
EXAMPLE:
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])
True
sage: p = x*B[0]
sage: is_linearly_dependent(B + [p])
False
sage: is_linearly_dependent([])
False
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 [Laz92].
INPUT:
OUTPUT:
True if the basis is triangular; False otherwise.
EXAMPLE:
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
Assuming that p is a linear combination of polys, determines coefficients that describe the linear combination. This probably doesn’t 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:
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].
EXAMPLE:
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]
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 [Laz92]. It is not necessary for the term ordering to be lexicographic.
INPUT:
OUTPUT:
A list T of triangular sets T_0, T_1, etc.
EXAMPLE:
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(), proof=True)
...
NotImplementedError: proof = True factorization not implemented. Call factor with proof=False.
sage: triangular_factorization(I.groebner_basis(), proof=False)
[[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]]
Educational Versions of Groebner Basis Algorithms.
Educational version of the 
-Groebner Basis Algorithm over PIDs.