The purpose of my talk is to present old and new results which lie on the border between Linear Algebra and Computational Commutative Algebra. The main source is the recent book Computational Linear and Commutative Algebra by Martin Kreuzer and myself. The talk starts by recalling some old results related to one endomorphism of a finitely generated vector space. Then we progress to considering families of pairwise commuting endomorphisms: this opens up a new world. The merits of this modern approach to linear algebra will be described. Among them, we will see the connection to many fundamental problems in computer algebra such as computing the primary decomposition of zero-dimensional ideals, and solving systems of polynomial equations. The final part of the talk will be devoted to exhibiting a link between advanced tools in linear algebra and an old theme in algebraic geometry.

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, "ElimTH", has as main step the computation of an elimination ideal via a \textit{truncated, homogeneous} Gröbner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm. Either may be used inside the third algorithm, "RatPar", to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm "ModImplicit", for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

CoCoA is a special-purpose system for doing Computations in Commutative Algebra. It runs on all common platforms.CoCoA's particular strengths include ideal/module operations (such as Gröbner bases, syzygies and minimal free resolutions, intersections, divisions, the radical of an ideal, etc), polynomial factorization, exact linear algebra, computing Hilbert functions, and computing with zero-dimensional schemes and toric ideals.The usefulness of these technical skills is enhanced by the mathematically natural language for describing computations. This language is readily learned by students, and enables researchers to explore and develop new algorithms without the administrative tedium necessary when using "low-level" languages.Lately the CoCoA project has entered a new phase: the new design is expressly developed as a C++ library; a server and a standalone interactive system will be built on top of this library. The design should reflect the underlying mathematical structure since this will ensure that the library is natural to use.In this tutorial we will show several applications of Computer Commutative Algebra through the use of CoCoA and CoCoALib.

In this paper we present a full solution to an open problem in Design of Experiments, a branch of Statistics, which can equally be seen as a problem in Algebraic Geometry. Given a complete set O of estimable terms, we are able to find all the fractions F of a full factorial design, such that O is a basis of P/I(F) as a K-vector space. This fact can be rephrased as a result in the theory of zem-dimensional schemes.

In this paper we use Grobner basis theory and some methods of Algebraic Geometry to solve a relevant problem in Statistics, more specifically in the Design of Experiments.Namely suppose we are given a Fhll Factorial Design D and a complete polynomial model P, whose support is contained in the order ideal of monomials defined by D. We show how to construct families of ideals defining Fractions ~of D which are minimally identified by P.

In this paper we continue the work done in [GMRT].Namely we describe an algorithm which allows to drive the classical Buchberger Algorithm by meansof multigrade Hilbert-Poincar6 series.The re-suIt is that in the multigrade caseone can kill many more useless critical pairs.Besides, the new algorithm, called MGHDriven, is intrinsically parallelizable and it is now implemented in ~~(see [cNR]).

In this paper redescribe some experimentti findings on se-