Teo Mora

Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering

We present an efficient algorithm for the transformation of a Grobner basis of a zero-dimensional ideal with respect to any given ordering into a Grobner basis with respect to any other ordering. This algorithm is polynomial in the degree of the ideal. In particular the lexicographical Grobner basis can be obtained by applying this algorithm after a total degree Grobner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchbergers algorithm.

Gröbner bases of ideals defined by functionals with an application to ideals of projective points

In this paper we study 0-dimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morphisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, generalizing the Buchberger-Möller algorithm for computing a basis of an ideal vanishing at a set of points and the FGLM basis conversion algorithm.As an application to Algebraic Geometry, we show how to compute in polynomial time a minimal basis of an ideal of projective points.

“One sugar cube, please” or selection strategies in the Buchberger algorithm

In this paper redescribe some experimentti findings on selection strategies for Gr6bner basis computation with the Buchberger algorithm. In particular, the results suggest that the “sugar flavor” of the “normal selection”, implemented first in COCOA, then in AlPI, [14], [15] (up to now in the muLISP version, in a short time in the COMMON-LISP version, including the parallel version, [1]) and now in SCRATCHPAD-II, is the best choice for a selection strategy. It has to be combined with the “straightforward” simplification strategy and with a special form of the Gebauer-Moller criteria to obtain the best results. The idea of the “sugar flavor” is the following: the Buchberger algorithm for homogeneous ideals, with degreecompatible term ordering and normal selection strategy, usually works fine. Homogenizing the basis of the ideal is good for the strategy, but bad for the basis to be computed. The sugar flavor computes, for every polynomial in the course of the algorithm, ‘(the degree that it would have if computed with the homogeneous algorithm”, and uses this phantom degree (the sugar) only for the selection strategy. We have tested several examples with different selection strategies, and the sugar flavor has proved to be always the best choice or very near to it. The comparison between the different variants of the sugar flavor has been made, but the results are up to now inconclusive. We include a complete deterministic description of the Buchberger algorithm as it was used in our experiments.l

Gröbner bases computation using syzygies

Bases Computation Mollert Teo Mora

The shape of the Shape Lemma

Using Syzygies*

On multiplicities in polynomial system solving

The Shape Lemma was originally introduced in [3] and so christened by Lakshman ([5]). It is an easy generalization of the Primitive Element Theorem and it states that a Odimensional radical ideal in a polynomial ring k[X1, . . . . Xn], after most changes of coordinates, has a basis {91(X1),X2 -92(X1 ),. ~.,xn -gn(xl)} Notwithstanding its triviality, it has proved ubiquitous in recent papers on polynomial system solving ([1, 2, 4, 6, 7]). The obvious example (X2, XY, Y2) is sufficient to show that some assumption is needed on a O-dimensional ideal in order that it holds; the obvious example (X2, Y) is sufficient to show that radicality is too strong an assumption. Since most of the results making use of the Shape Lemma are valid whenever the Shape Lemma holds and are of interest also for non radical ideals, it is worthwhile to exactly characterize those O-dimensional ideals to which the Shape Lemma applies. It turns out that this exact characterization is as trivial as the original Shape Lemma itself. In fact both this characterization and the generalization of it we give are easy specializations of a classical result in algebraic geometry on the minimum dimension of a generic biregular projection of a variety as a function of its dimension and of the dimension of its tangent bundle. We give a direct, elementary, self-contained proof of this specialization.

Grbner Bases, Coding, and Cryptography

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

Gröbner bases of ideals given by dual bases

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Grbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations. This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation.

Decoding Cyclic Codes: the Cooper Philosophy

INTRODUCTION In 1982, Buchberger and Moller proposed an algorithm which, given a finite number of rational points in the affme ndimensional space, computes a Grobner basis for the ideal I of the polynomials vanishing at the points. In 1988, Faugere, Giarmi, Lazard and Mora supplied an algorithm, which given the reduced Grobner basis w.r.t. some term-ordering <1 of a O-dim. ideal I, returns its reduced Grdbner basis w.r.t. some other term-ordering <2. Both algorithms (and more recent variants of the FGLM algorithm) share the same structure: an explicit K-vector space morphism v : KIX1,..., Xn] + Ks is given with kernel the ideal I; therefore it is possible to compute linear dependence relations mod. I among terms in KIX1,... ,Xn], leading to the

Gröbner duality and multiplicities in polynomial system solving

In 1990 Cooper suggested to use Grobner basis computations in order to deduce error locator polynomials of cyclic codes.