Anna Maria Bigatti

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.

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.

Generic Initial Ideals and Distractions

ABSTRACT The generic initial ideals of a given ideal are rather recent invariants. Not much is known about these objects,and it turns out to be very difficult to compute them. The main purpose of this paper is to study the behaviour of generic initial ideals with respect to the operation of taking distractions. Theorem 4.3 is our main result. It states that the DegRevLex-generic initial ideal of the distraction of a strongly stable ideal is the ideal itself. In proving this fact, we develop some new results related to distractions, stable and strongly stable ideals. We draw some geometric conclusions for ideals of points.

On the computation of hilbert—Poincaré series

We prove a theorem, which provides a formula for the computation of the Poincaré series of a monomial ideal ink[X1,⋯, Xn], via the computation of the Poincaré series of some monomial ideals ink[X1,⋯, Xi,⋯, Xn]. The complexity of our algorithm is optimal for Borel-normed ideals and an implementation in CoCoA strongly confirms its efficiency. An easy extension computes the Poincaré series of graded modules over standard algebras.

Computing inhomogeneous Gröbner bases

In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Grobner bases via Buchbergers Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use as a main tool the procedure of a self-saturating Buchbergers Algorithm. Another strictly related topic is treated later when a mathematical foundation is given to the sugar trick which is nowadays widely used in most of the implementations of Buchbergers Algorithm. A special emphasis is also given to the case of a single grading, and subsequently some timings and indicators showing the practical merits of our approach.

Borel sets and sectional matrices

Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial structure of Borel ideals. This enables us to prove theorems on the shape of thesectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function.

SC2 : Satisfiability Checking Meets Symbolic Computation

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted \(\mathsf {SC}^\mathsf{2} \) project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified \(\mathsf {SC}^\mathsf{2} \) community.Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted SC 2 project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified SC 2 community.

Monomial ideals, computations and applications

A survey on Stanley depth.- Stanley decompositions using CoCoA.- A beginners guide to edge and cover ideals.- Edge ideals using Macaulay2.- Local cohomology modules supported on monomial ideals.- Local Cohomology using Macaulay2.

SC2: Satisfiability Checking Meets Symbolic Computation (Project Paper)

Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted \(\mathsf {SC}^\mathsf{2} \) project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified \(\mathsf {SC}^\mathsf{2} \) community.Symbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted SC 2 project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified SC 2 community.

CoCoA: computations in commutative algebra

CoCoA is a special-purpose system for doing Computations in Commutative Algebra. It runs on all common platforms.