Short interval results for powerfree polynomials over finite fields

Abstract

Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.