Generic Newton polygon for exponential sums in $n$ variables with parallelotope base
Abstract
Let $p$ be a prime number. Every $n$variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an ArtinSchreierWitt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$function associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is an $n$dimensional paralleltope $\Delta$. We denote this polygon by $\mathrm{GNP}(\Delta)$. We prove a lower bound of $\mathrm{GNP}(\Delta)$, which is called the improved Hodge polygon $\mathrm{IHP}(\Delta)$. We show that $\mathrm{IHP}(\Delta)$ lies above the usual Hodge polygon $\mathrm{HP}(\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with $\mathrm{GNP}(\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(\Delta)$.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1710.00137
 Bibcode:
 2017arXiv171000137R
 Keywords:

 Mathematics  Number Theory
 EPrint:
 36 pages