Square-full polynomials in short intervals and in arithmetic progressions

Abstract

We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_q[T]$ of polynomials over a finite field $F_q$ of $q$ elements, in the limit $q\to \infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.