2012/07/30 by Antonio Campillo, Campillo, Antonio, Ann Lemahieu +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1207.7057
openalex publication_date 2012/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To study singularities on complex varieties we study Poincar 'e series of\nfiltrations that are defined by discrete valuations on the local ring at the\nsingularity. In all previous papers on this topic one poses restrictions on the\ncenters of these valuations and often one uses several definitions for\nPoincar 'e series. In this article we show that these definitions can differ\nwhen the centers of the valuations are not zero-dimensional, i.e. do not have\nthe maximal ideal as center. We give a unifying definition for Poincar 'e\nseries which also allows filtrations defined by valuations that are all\nnonzero-dimensional. We then show that this definition satisfies a nice\nrelation between Poincar 'e series for embedded filtrations and Poincar 'e\nseries for the ambient space and we give some application for singularities\nwhich are nondegenerate with respect to their Newton polyhedron.\n