## Local Zeta Functions for Non-degenerate Laurent Polynomials Over p-adic Fields

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 4, Page 569–595.

León-Cardenal, E. ; Zúñiga-Galindo, W. A.
Local Zeta Functions for Non-degenerate Laurent Polynomials Over p-adic Fields

Abstract:
In this article, we study local zeta functions attached to Laurent polynomials over $p$-adic fields, which are non-degenerate with respect to their Newton polytopes at infinity. As an application we obtain asymptotic expansions for $p$-adic oscillatory integrals attached to Laurent polynomials. We show the existence of two different asymptotic expansions for $p$-adic oscillatory integrals, one when the absolute value of the parameter approaches infinity, the other when the absolute value of the parameter approaches zero. These two asymptotic expansions are controlled by the poles of twisted local zeta functions of Igusa type.

Keywords: $p$-adic oscillatory integrals, Laurent polynomials, Igusa zeta function, Newton polytopes, non-degeneracy conditions at infinity.

Mathematics Subject Classification (2010): Primary 14G10, 11S40; Secondary 11T23, 14M25.
Mathematical Reviews Number: MR3185292