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On the Poles of Maximal Order of the Topological Zeta Function
Author(s) -
Laeremans Ann,
Veys Willem
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398005566
Subject(s) - mathematics , polyhedron , order (exchange) , riemann zeta function , polynomial , topology (electrical circuits) , function (biology) , polylogarithm , degenerate energy levels , rational function , pure mathematics , singularity , germ , arithmetic zeta function , combinatorics , prime zeta function , mathematical analysis , finance , evolutionary biology , economics , biology , physics , quantum mechanics
The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non‐degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/ N with N a positive integer. 1991 Mathematics Subject Classification 14B05, 14E15, 32S50.