A sup-Hodge bound for exponential sums

TL;DR

This paper establishes an explicit arithmetic bound for the Newton polygon of the $C$-function of $T$-adic exponential sums, showing it lies above the Hodge polygon, thus providing a sup-Hodge bound for $p$-power order exponential sums.

Contribution

It introduces a new explicit bound for the Newton polygon of the $C$-function, linking it to the Hodge polygon and advancing understanding of exponential sums.

Findings

Newton polygon of the $C$-function lies above the Hodge polygon

Established a sup-Hodge bound for $p$-power order exponential sums

Provides explicit arithmetic bounds for $T$-adic exponential sums

Abstract

The $C$-function of $T$-adic exponential sums is studeid. An explicit arithmetic bound is established for the Newton polygon of the $C$-function. This polygon lies above the Hodge polygon. It gives a sup-Hodge bound of the $C$-function of $p$-power order exponential sums.