Generic Newton polygon for exponential sums in $n$ variables with parallelotope base

TL;DR

This paper establishes a lower bound for the Newton polygon of L-functions associated with generic polynomials over finite fields, revealing its relation to the Hodge polygon and providing insights into the distribution of slopes.

Contribution

It introduces the improved Hodge polygon as a lower bound for the Newton polygon and shows its coincidence with the Newton polygon for large primes, advancing understanding of exponential sums in multiple variables.

Findings

The improved Hodge polygon bounds the Newton polygon from below.

For sufficiently large primes, the improved Hodge polygon matches the Newton polygon at infinitely many points.

The distribution of slopes of the Newton polygon is roughly characterized.

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 $\Delta$. 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 $\mathrm{GNP}(\Delta)$.