Short Interval Results For Powerfree Polynomials Over Finite Fields

TL;DR

This paper extends the understanding of the distribution of powerfree polynomials over finite fields in short intervals, generalizing previous results and developing polynomial analogues of classical number theory techniques.

Contribution

It generalizes recent theorems on squarefree polynomials to all k-free polynomials and develops polynomial versions of classical techniques for studying gaps.

Findings

Distribution results for k-free polynomials in short intervals

Generalization of Carmon and Entin's theorem to all k≥2

Polynomial analogues of classical number theory results

Abstract

Let $k \geq 2$ 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.