On the Moments of Exponential Sums over r-Free Polynomials

TL;DR

This paper determines the exact order of magnitude of moments of exponential sums over r-free polynomials in finite fields, extending techniques from integer settings and employing a function field Hardy-Littlewood circle method.

Contribution

It provides the first precise asymptotic formulas for moments of exponential sums over r-free polynomials in finite fields for all positive k.

Findings

Exact order of magnitude for moments of exponential sums.

Asymptotic formulas in the supercritical case k>1+1/r.

Extension of integer techniques to function fields.

Abstract

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over the finite field $\mathbb{F}_q$. Building off of techniques of Balog and Ruzsa and of Keil in the integer setting, we determine the precise order of magnitude of $k$th moments of exponential sums over $r$-free polynomials in $\mathbb{F}_q[t]$ for all $k>0$. In the supercritical case $k>1+1/r$, we acquire an asymptotic formula using a function field analogue of the Hardy-Littlewood circle method.