Bulletin of the London Mathematical Society



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ON THE POLES OF IGUSA'S LOCAL ZETA FUNCTION FOR ALGEBRAIC SETS


W. A. ZUNIGA-GALINDO a1
a1 Barry University, Department of Mathematics and Computer Science, 11300 N.E. Second Avenue, Miami Shores, FL 33161, USA wzuniga@mail.barry.edu

Article author query
zuniga-galindo w   [Google Scholar] 
 

Abstract

Let $K$ be a $p$-adic field, let $Z_{\Phi }(s,f)$, $s\,{\in}\,\mathbb{C}$, with Re$(s)\,{>}\,0$, be the Igusa local zeta function associated to $f(x)\,{=}\,(f_{1}(x),\ldots,f_{l}(x))\,{\in}\,[ K( x_{1},\ldots,x_{n})]^{l}$, and let $\Phi $ be a Schwartz–Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of $Z_{\Phi }(s,f)$. Using resolution of singularities it is possible to express $Z_{\Phi }(s,f)$ as a finite sum of $p$-adic monomial integrals. These monomial integrals are computed explicitly by using techniques of toroidal geometry. In this way, an explicit list of the candidates for poles of $Z_{\Phi }(s,f)$ is obtained.

(Received March 25 2002)
(Revised April 17 2003)

Maths Classification

11S40; 14M25; 11D79.