Short Interval Results For Powerfree Polynomials Over Finite Fields

2023/10/03 by Angel Kumchev, Kumchev, Angel, Nathan McNew +3
Computer Science · Mathematics · #11T55 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.02495

openalex publication_date 2023/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k ≥ 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 kth 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 ≥ 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.

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