On the Moments of Exponential Sums over r-Free Polynomials

Ben Doyle

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arXiv · 2506.20581

On the Moments of Exponential Sums over r-Free Polynomials

Ben Doyle

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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.

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Ben Doyle. 2025-06-25. On the Moments of Exponential Sums over r-Free Polynomials. https://arxiv.org/abs/2506.20581

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