Poles of Archimedean zeta functions for analytic mappings

Abstract

Let f = (f1, . . . , f l ) : U → K l , with K = R or C, be a K-analytic mapping defined on an open set U ⊂ K n , and let Φ be a smooth function on U with compact support. In this paper, we give a description of the possible poles of the local zeta function attached to (f , Φ) in terms of a log-principalization of the ideal I f = (f1, . . . , f l ). When f is a non-degenerate mapping, we give an explicit list for the possible poles of Z Φ(s, f ) in terms of the normal vectors to the supporting hyperplanes of a Newton polyhedron attached to f , and some additional vectors (or rays) that appear in the construction of a simplicial conical subdivision of the first orthant. These results extend the corresponding results of Varchenko to the case l 1, and K = R or C. In the case l = 1 and K = R, Denef and Sargos proved that the candidate poles induced by the extra rays required in the construction of a simplicial conical subdivision can be discarded from the list of candidate poles. We extend the Denef-Sargos result to arbitrary l 1. This yields, in general, a much shorter list of candidate poles, which can, moreover, be read off immediately from Γ(f ).wheref (y) := (f 1 (y), . . . ,f l (y)), in V .We can choose a small neighbourhood V b of b over which (2.3)-(2.5) are valid, and |ε(y)| K , |f (y)| K , |η(y)| K are R-analytic. Then |ε(y)| s K , |f (y)| s K are R-analytic in y for any s ∈ C, and holomorphic in s ∈ C for any y ∈ V b .Since h −1 (support Φ) is compact, we can take a finite covering of the form {V b }, where the V b are homeomorphic under φ V to the polydisc P (0) in K n defined by |y i | K < , with sufficiently small and for 1 i n. By picking a smooth partition of the unity subordinate to {V b }, and using the previous discussion,becomes a finite sum of integrals of the following two types:where Ψ is a C ∞ 0 function with support contained in a polydisc P (0) and e s ln |f * (y)|K is R-analytic for y ∈ V b and holomorphic in s ∈ C, or J(s) := K n Θ(y, s)

On Archimedean zeta functions and Newton polyhedra

“…Perhaps our methods, combined with those of [16] may give a better estimation and/or asymptotic expansions of complex oscillatory integrals like the ones studied in [33]. The subject of real oscillatory integrals constitutes a very active area of research, see (among many others) [6,7,8,11,15,16,17,18,19,23,24,25,26,29,31,32,33,34]. We believe that our results may be of some interest in this community.…”

“…In the case K = R, Denef and Sargos proved in [11] that the poles coming from those extra rays can be discarded, thus reducing the list of candidate poles. In [29] this proof is extended for real zeta functions of analytic mappings. In this work we prove the analogue of the results of Denef and Sargos for the case K = C, thus providing a much shorter list of candidate poles that can be read off directly from the geometry of the Newton polyhedron of f .…”

Archimedean Zeta Functions and Oscillatory Integrals

“…Another local zeta function involving several functions was studied in a joint work with W. Veys and W. A. Zúñiga-Galindo [54], were we considered Archimedean zeta functions for analytic mappings. If K = R or C, let f = (f 1 , .…”

An explicit formula for the local zeta function of a Laurent polynomial

“…In particular, after the pioneering work of Varchenko [9], Newton polyhedra techniques have been extensively employed to study local zeta functions as well as their connections with oscillatory integrals, see e.g. [1,7] and the references therein for the Archimedean case, and [2,10,12], among others, in the non-Archimedean case, including the positive characteristic case. In [5,6] we began the study of local zeta functions for a Laurent polynomial f over a p−adic field.…”