Résume | In his famous proof of the existence of a resolution of singularities in characteristic zero (1964) Hironaka also introduced several constructive/algorithmic ideas, most prominently the notion of standard bases which have become important tools in computational algebraic geometry nowadays. Other constructive aspects, however, seemed to be without further practical applications.
In this talk I shall explain solutions to two tasks of very different flavour in computational algebraic geometry, which profit each in their own way from Hironaka's work: A massively parallel algorithm to decide non-singularity of a variety arises from the termination criterion of desingularization and makes use of descent in ambient dimension by means of hypersurfaces of maximal contact. On the other hand, when counting subrings (in the sense of order zeta-functions) p-adic integrals arise for which the domain of integration seems rather inaccessible at first glance. However, a suitable desingularization transforms the task into a number of easier problems each of which can be tackled by standard methods. |