A normal form algorithm for the Brieskorn lattice

This article describes a normal form algorithm for the Brieskorn lattice of an isolated hypersurface singularity. It is the basis of efficient algorithms to compute the Bernstein-Sato polynomial, the complex monodromy, and Hodge-theoretic invariants of the singularity such as the spectral pairs and good bases of the Brieskorn lattice. The algorithm is a variant of Buchberger's normal form algorithm for power series rings using the idea of partial standard bases and adic convergence replacing termination.Comment: 23 pages, 1 figure, 4 table

Invariantes analíticos de singularidades aisladas de hipersuperficie e invariantes combinatorios de semigrupos numéricos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 11-07-2022This 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...Este trabajo trata sobre invariantes analíticos de singularidades aisladas de hipersuperficie e invariantes combinatorios de semigrupos numéricos. La primera parte trata sobre invariantes analíticos y topológicos de una singularidad aislada de hipersuperficie. Nuestras principales contribuciones son las siguientes: primero proporcionamos una formula cerrada para el numero mínimo de Tjurina en una clase de equisingularidad de una rama plana en términos de invariantes topológicos de la rama, segundo abordamos una pregunta de Dimca y Greuel sobre el cociente del numero de Milnor y del numero de Tjurina de una singularidad aislada de curva plana; extendemos esta pregunta a singularidades aisladas de super cie en C3; lo cual es clave para proporcionar una respuesta completa a la pregunta de Dimca y Greuel. Ademas, mostramos la conexión de esta extensión con una conjetura planteada por Durfee. Finalmente, establecemos la distribución continua límite de K. Saito para el espectro de singularidades aisladas de hipersuperficie Newton no degeneradas y vinculamos este problema con nuestra generalización de la pregunta de Dimca y Greuel. Como consecuencia, esto proporciona una nueva forma de entender la importancia de la conjetura de Durfee en el contexto de las singularidades aisladas de hipersuperficie..Fac. de Ciencias MatemáticasTRUEunpu

Algorithms for the Gauss–Manin connection

We give an introduction to the theory of the Gauss-Manin connection of an isolated hypersurface singularity and describe an algorithm to compute the V-filtration on the Brieskorn lattice. We use an implementation in the computer algebra system Singular to prove C. Hertling’s conjecture about the variance of the spectrum for Milnor number µ ≤ 16