Three Algorithms in Algebraic Geometry, Coding Theory and Singularity Theory

Abstract

We describe three algorithms in algebraic geometry, coding theory and singularity theory, which are new, resp. have new ingredients. Moreover, we put special emphasis on their implementation in the computer algebra system Singular. The first algorithm computes 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, of bases of adjoint forms, and of the linear system of a given rational divisor on the normalization of the curve. Finally, the third algorithm provides a method to compute the V-filtration, the monodromy and the singularity spectrum of an arbitrary isolated hypersurface singularity.