2004/05/17 by Jan Denef, Johannes Nicaise, Denef, Jan +3
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.CA
paper · pdf · doi:10.48550/arxiv.math/0405317
21 pages
arxiv created 2004/05/17 · openalex publication_date 2004/05/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton polyhedron. In particular, up to an easy non-zero factor, this residue only depends on the (tau0)-principal part of f, where tau0 is the smallest face of the Newton polyhedron intersecting the diagonal. This formula allows us to prove some vanishing results for the residue. More precisely, we prove that the residue vanishes when tau0 is unstable, and we give a partial proof of the reverse implication in the complex case. We also deduce an explicit formula for the residue, in the case where tau0 is a simplex of codimension 1, and the only points of the support of f on tau0 are its vertices.