Short Interval Results For Powerfree Polynomials Over Finite Fields

Abstract

Let $k\ge 2$ be an integer and ${\mathbb{F}}_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in ${\mathbb{F}}_q[x]$ that are not divisible by the $k$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 $k\ge 2$. We also develop polynomial versions of the classical techniques used to study gaps between $k$-free integers in $\mathbb{Z}$. We apply these techniques to obtain analogues in ${\mathbb{F}}_q[x]$ of some classical theorems on the distribution of $k$-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.