The $p$-powers dividing certain exponential sums

TL;DR

This paper introduces a new concept called couple density to analyze the $p$-powers dividing exponential sums, establishing bounds related to the $q$-adic valuation of associated $L$-functions.

Contribution

It defines couple density and determines a minimum bound for the $q$-adic valuation of exponential sums, linking it to zeros and poles of $L$-functions.

Findings

Established a minimum bound for the $q$-adic valuation based on couple density.

Linked the valuation bounds to zeros and poles of the associated $L$-function.

Provided a framework for analyzing divisibility properties of exponential sums.

Abstract

We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms of the density of the couple $(D,\mathbf b)$ for the $q$-adic valuation of the sum $ S_{\ell}(F,\mathbf b)$ with $F$ a Laurent polynomial. And we show that this minimum is a bound for the $q$-adic valuation of the zeros and poles of the associated $L$-function.