Given a zero-dimensional ideal I in a polynomial ring, many algorithms start by finding univariate polynomials in~I, or by computing a lex-Groebner basis of~I. These are related to considering the minimal polynomial of an element in P/I, which may be computed using Linear Algebra from a Groebner Basis (for any term-ordering). In this tutorial we'll see algorithms for computing minimal polynomials, applications of modular methods, and then some applications, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality. We'll also address a kind of opposite problem: given a "geometrical description'', such as a finite set of points, find the ideal of polynomials which vanish at it. We start from the original Buchberger-Moeller algorithm, and we show some developments. All this will be done with a special eye on the practical implementations, and with demostrations in CoCoA.

We propose algorithms based on monomial resolution theory for simplicial homology computation. We explore some alternatives that can either be used as a preprocessing step for homology computation or as alternatives to the usual linear algebra approach. We show the results of some computer experiments to demonstrate the performance of a C++ implementation using the computer algebra library CoCoALib.

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.

Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n - 1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions. This feature of (n - 1)-st Koszul homology allows us to transform an algorithm that computes Koszul homology of monomial ideals to use it for the computation of irreducible and Stanley decompositions. This is an example of how algorithms and structures specifically targeted to computations on monomial ideals should take into account the combinatorial properties of them to produce efficient methods, an issue that is worth introducing into modern computer algebra systems. To illustrate this fact we present some details on the implementation of the algorithm in CoCoALib.

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.