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 $\mathbb{Z}_p$. 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 $Δ$. We denote this polygon by $\mathrm{GNP}(Δ)$. We prove a lower bound of $\mathrm{GNP}(Δ)$, which is called the improved Hodge polygon $\mathrm{IHP}(Δ)$. We show that $\mathrm{IHP}(Δ)$ lies above the usual Hodge polygon $\mathrm{HP}(Δ)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $Δ$, it coincides with $\mathrm{GNP}(Δ)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(Δ)$.