Igusa's Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials
Abstract
We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Z\'u\~niga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.
Keywords
Cite
@article{arxiv.1604.02497,
title = {Igusa's Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials},
author = {Adriana A. Albarracin-Mantilla and Edwin León-Cardenal},
journal= {arXiv preprint arXiv:1604.02497},
year = {2017}
}
Comments
20 pages. In this version there is a more precise statement of Lemma 2.4 and a correction to the Example in Section 4. Minor corrections added