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# Introduction

Algorithmic and computational aspects have become a major, still growing issue in mathematical research and teaching. This has various deeper reasons in the cultural and technological development of today's society but also quite practical reasons. One of these is certainly the existence and maturity of software systems which, via implemented algorithms, provide easy access to hard and sometimes sophisticated computations, assisting and supporting mathematical research.

In this article, we describe three algorithms in algebraic geometry, coding theory and singularity theory, which are new, resp. have new ingredients. The first one describes how to compute the normalization of an affine reduced ring, an ideal defining the non-normal locus and, as an application, the integral closure of an ideal. The second is devoted to the computation of the places of a projective plane curve defined over a finite field, and the computation of bases of adjoint forms and of the linear system of a given rational divisor on the normalization of the curve. Finally, the third one provides a method to compute the V-filtration, the monodromy and the singularity spectrum of an arbitrary isolated hypersurface singularity.

All three algorithms require non-trivial up to deep mathematical knowledge and go beyond foundational algorithms in computer algebra. Indeed, one of the purposes of this note is to show that highly complex mathematical objects can nowadays be represented in a computer and, thus, can be used in mathematical research on a higher level than ever before. All the algorithms described in this paper are implemented in the computer algebra system SINGULAR [17] as a free service to the mathematical community.

The normalization algorithm is based on an old criterion of Grauert and Remmert and has already been published [6,5]. From this it is not difficult to derive the principle for algorithms to compute the non-normal locus and the integral closure of an ideal, however, the concrete description and its realization appear to be new. The proposed algorithm for computing the places of plane curves is based on the Hamburger-Noether development and has been described on a theoretical level in [3], as well as the Brill-Noether algorithm for computing bases of linear systems. Since then, this algorithm has been implemented, together with a full coding and decoding algorithm, and we mainly concentrate on new algorithmic and computational aspects. It should be mentioned that the construction of AG codes, using quadratic transformations instead of Hamburger-Noether expansions, has been described and implemented before [28,19]. Finally, the algorithm to compute the V-filtration and the singularity spectrum is very recent, a theoretical description of parts of it are to be published in [37]. Here, we give a short description of the theory together with a description of some computational aspects. A paper with full details will be published by the third author.

Next: Integral closure of rings Up: Three Algorithms in Algebraic Previous: Three Algorithms in Algebraic
Christoph Lossen
2001-03-21