Lorenzo Robbiano

“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

On the theory of graded structures

It is well known that various notions of distinguished bases of ideals, such as standard and Grobner bases, and algorithms for constructing them, play a central role in solving algorithmic problems in polynomial ideal theory and related algebraic theories. In this paper a new structural algebraic framework is given for the concept and the construction of the so-called generalised standard bases. Although the spirit of this paper is more structural than algorithmic, the general results achieved should also help to shape and direct future research in the algorithmic aspects of commutative algebra.

Computing Ideals of Points

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger?Moller algorithm, best suited for the computation over Q, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in CoCoA, and we report some experimental timings.

Cayley-Bacharach schemes and their canonical modules

A set of s points in P d is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s − 1 points has the same Hilbert function. We investigate the consequences of this «weak uniformity.» The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB-scheme X has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points

Computing Toric Ideals

Toric ideals are binomial ideals which represent the algebraic relations of sets of power products. They appear in many problems arising from different branches of mathematics. In this paper, we develop new theories which allow us to devise a parallel algorithm and an efficient elimination algorithm. In many respects they improve existing algorithms for the computation of toric ideals.

An algebraist’s view on border bases

This chapter is devoted to laying the algebraic foundations for border bases of ideals. Using an order ideal \(\mathcal{O}\), we describe a zero-dimensional ideal from the outside. The first and higher borders of \(\mathcal{O}\) can be used to measure the distance of a term from \(\mathcal{O}\) and to define \(\mathcal{O}\)-border bases. We study their existence and uniqueness, their relation to Grobner bases, and their characterization in terms of commuting matrices. Finally, we use border bases to solve a problem coming from statistics.

On set-theoretic complete intersections in the projective space

SuntoIn questo lavoro si considerano molte classi di varietà proiettive che sono insiemisticamente intersezione completa. Utilizzando la teoria delle basi di Gröbner, si deduce un metodo unificante di verifica della suddetta proprietà.SummaryIn this paper we consider many classes of projective varieties which are set-theoretically complete intersections; by using the theory of the Gröbner bases, we deduce an unifying method for the verification of such property.

Efficiently computing minimal sets of critical pairs

Abstract In the computation of a Grobner basis using Buchberger’s algorithm, a key issue for improving the efficiency is to produce techniques for avoiding as many unnecessary critical pairs as possible. A good solution would be to avoid all non-minimal critical pairs, and hence to process only a minimal set of generators of the module generated by the critical syzygies. In this paper we show how to obtain that desired solution in the homogeneous case while retaining the same efficiency as with the classical implementation. As a consequence, we get a new optimized Buchberger algorithm.

Number of generators of ideals

Let I be a homogeneous ideal of a polynomial ring over a field, v(I ) the number of elements of any minimal basis of I, e = e(I ) the multiplicity or degree of R/I, h = h(I ) the height or codimension of I, i = indeg ( I ) the initial degree of J , i.e. the minimal degree of non zero elements of I . This paper is mainly devoted to find bounds for v(I ) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.

On border basis and Gröbner basis schemes

Hilbert schemes of zerodimensional ideals in a polynomial ring can be covered with suitable affine open subschemeswhose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. and in this situation they show a clear advantage with respect to Gröbner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications.In this paper we compare Gröbner basis schemes with border basis schemes. It is shown that Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof thatall the ideals which define a Gröbner basis scheme and are obtained using Buchbergers Algorithm, are equal. Another result is that if the origin (i.e. the point corresponding to the unique monomial ideal) in the Gröbner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Gröbner basis schemes. Since it is natural to look for situations where a Gröbner basis scheme and the corresponding border basis scheme are equal, we address the issue, provide an answer, and exhibit some consequences. Open problems are discussed at the end of the paper.