Carlo Traverso

“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

Hilbert functions and the Buchberger algorithm

Abstract In this paper we show how to use the knowledge of the Hilbert–Poincare series of an ideal I to speed up the Buchberger algorithm for the computation of a Grobner basis. The algorithm is useful in the change of ordering and in the validation of modular computations, also with tangent cone orderings; speeds the direct computation of a Grobner basis if the ideal is a complete intersection, e.g. in the computation of cartesian from parametric equations, can validate or disprove a conjecture that an ideal is a complete intersection, and is marginally useful also when the conjecture is false. A large set of experiments is reported.

Gröbner bases computation using syzygies

Bases Computation Mollert Teo Mora

The shape of the Shape Lemma

Using Syzygies*

Numerical stability and stabilization of Groebner basis computation

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.

Strategy-accurate parallel Buchberger algorithms

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known.We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms.We suggest some algorithms to deal with the case of overdetermined and unstable systems.

Grbner Bases, Coding, and Cryptography

Abstract We describe two parallel versions of the Buchberger algorithm for computing Grobner bases, one for the general case and one for homogeneous ideals, which exploit coarse grain parallelism. For the general case, to avoid the growth in number and complexity of the polynomials to reduce, the algorithm adheres strictly to the same strategies as the best sequential implementation. A suitable communication procotol has been designed to ensure proper synchronization of the various processes and to limit their idle time. We provide a detailed analysis the maximum potential degree of parallelism that is achievable with such architecture. The analysis corresponds to the results of our experimental implementation and also explains similar results obtained by other authors.

Yet Another Ideal Decomposition Algorithm

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.

Algorithms to compute the topology of orientable real algebraic surfaces

The problem of decomposing an ideal into pure-dimensional components is a key step in several basic algorithms of commutative algebra. In this paper we describe algorithms for equidimensional decompositions, that completely avoid generic or random projections, and does not need lexicographic Grobner bases or characteristic sets, hence can be a candidate to a best competitor against direct algorithms.

Newton Symmetric Functions and the Arithmetic of Algebraically Closed Fields

We present constructive algorithms to determine the topological type of a non-singular orientable real algebraic projective surface S in the real projective space, starting from a polynomial equation with rational coefficients for S. We address this question when there exists a line in RP3 not intersecting the surface, which is a decidable problem; in the case of quartic surfaces, when this condition is always fulfilled, we give a procedure to find a line disjoint from the surface. Our algorithm computes the homology of the various connected components of the surface in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.