Formal Desingularization of Surfaces - The Jung Method Revisited -

In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there is evidence that the weaker concept is enough for many computational purposes. We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily algebraically closed. The paper includes a generalization of Duval's Theorem on rational Puiseux parametrizations to the multivariate case and a detailed description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure

A computational approach to the theory of adjoints

International audienceThe word "adjoint" refers to several definitions which are not all equivalent: we will deal with any of them. The aim of this work is to provide an algorithm which, given two plane curves D, H allows to decide whether H is adjoint to D. With a slight modification to the main procedure, we will be able to deal with special adjoints and true adjoints.Le mot "adjoint" fait référence à plusieurs définitions qui ne sont pas toutes équivalentes: nous en traiterons toutes. Le but de ce travail est de fournir un algorithme qui, étant donné les deux courbes planes D, H permet de décider si H est adjoint à D. Avec une légère modification de la procédure principale, nous pourrons traiter des ajoints spéciaux et les vrais adjoints

First Steps Towards Radical Parametrization of Algebraic Surfaces

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.Comment: 31 pages, 7 color figures. v2: added another case of genus

The Computation of the Logarithmic Cohomology for Plane Curves

We give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables. (Fixed typos of v1)Comment: 19 page