Invariantes analíticos de singularidades aisladas de hipersuperficie e invariantes combinatorios de semigrupos numéricosAnalytic invariants of isolated hypersurface singularities and combinatorial invariants of numerical semigroups

Abstract

This work is about analytic invariants of isolated hypersurface singularities and combinatorial invariants of numerical semigroups. The first part deals with analytic and topological invariants of an isolated hypersurface singularity. Our main contributions are the following: first we provide a closed formula for the minimal Tjurina number in an equisingularity class of a plane branch in terms of topological invariants of the branch, secondly we address a question of Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of an isolated plane curve singularity; we extend this question to isolated surface singularities in C3 which gives the clue to provide a complete answer to Dimca and Greuel's question. Moreover, we show the connection of the extended question with an old standing conjecture posed by Durfee. Finally, we establish K. Saito's continuous limit distribution for the spectrum of Newton non-degenerate isolated hypersurface singularities and link this problem with our generalization of Dimca and Greuel's question. As a consequence, this provides a new way of understanding the important role of Durfee's conjecture in the context of isolated hypersurface singularities...