On the Moments of Exponential Sums over r-Free Polynomials

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 $k>1+1\mathrm{/}r$, we acquire an asymptotic formula using a function field analogue of the Hardy-Littlewood circle method.