Poles of the topological zeta function for plane curves and Newton polyhedra

Ann Lemahieu (1) , Lise van Proeyen (2)

1 UNS - Université Nice Sophia Antipolis (1965 - 2019)
2 KU Leuven - Catholic University of Leuven = Katholieke Universiteit Leuven

The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For functions that are nondegenerate with respect to their Newton polyhedron it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, W. Veys showed how to ¯lter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles.

Domaines: Mathématiques [math]

Ann Lemahieu, Lise van Proeyen. Poles of the topological zeta function for plane curves and Newton polyhedra. Comptes Rendus. Mathématique, 2009, 347 (11), pp.637-642. ⟨10.1016/j.crma.2009.03.032⟩. ⟨hal-01280031⟩