Square-full polynomials in short intervals and in arithmetic progressions

TL;DR

This paper investigates the variance of square-full polynomial sums in short intervals and arithmetic progressions over finite fields, utilizing equidistribution results and matrix integrals to derive explicit formulas.

Contribution

It introduces a novel approach to compute variances of square-full polynomial sums using equidistribution and matrix integrals in the context of finite fields.

Findings

Derived explicit formulas for variances in the limit as q approaches infinity.

Connected variance calculations to triple matrix integrals over the unitary group.

Extended understanding of polynomial distribution in finite field 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 $q\rightarrow\infty$. 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.