L-functions of Certain Exponential Sums over Finite Fields

TL;DR

This paper fully characterizes the slopes and weights of L-functions associated with a class of exponential sums over finite fields, providing sharp estimates and counterexamples to existing conjectures.

Contribution

It offers a complete determination of slopes and weights for these L-functions, utilizing advanced tools like Adolphson-Sperber's work and Wan's decomposition theorems, and presents a counterexample to a conjecture.

Findings

Sharp estimate of exponential sums.

Explicit counterexample to Adolphson-Sperber's conjecture.

Complete determination of slopes and weights.

Abstract

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on weights of toric exponential sums.