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 Artin-Schreier-Witt tower of varieties whose Galois group is isomorphic to ${\Bbb Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a nontrivial finite character of ${\Bbb Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\Delta$. We denote this polygon by ${\rm GNP}(\Delta)$. We prove a lower bound of ${\rm GNP}(\Delta)$, which is called the improved Hodge polygon ${\rm IHP}(\Delta)$. We show that ${\rm IHP}(\Delta)$ lies above the usual Hodge polygon ${\rm HP}(\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with ${\rm GNP}(\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of ${\rm GNP}(\Delta)$.