$T$-adic exponential sums of polynomials in one variable

TL;DR

This paper investigates the properties of $T$-adic exponential sums of single-variable polynomials, establishing explicit bounds for their Newton polygons using polynomial exponents, which in turn inform bounds for related $L$-functions.

Contribution

It introduces an explicit arithmetic polygon based on the highest two exponents of the polynomial as a lower bound for the Newton polygon of the $C$-function, advancing understanding of $T$-adic exponential sums.

Findings

Established explicit lower bounds for Newton polygons

Connected bounds for $C$-function and $L$-function of exponential sums

Provided a new tool for analyzing $p$-adic exponential sums

Abstract

The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of exponential sums of $p$-power order.