Squarefree polynomials with prescribed coefficients

TL;DR

This paper proves that for large enough finite fields, one can select coefficients from specified subsets to construct squarefree polynomials, under certain size conditions of these subsets.

Contribution

It establishes the existence of squarefree polynomials with prescribed coefficients from subsets of finite fields, extending previous results to broader subset size conditions.

Findings

Existence of squarefree polynomials with prescribed coefficients in finite fields

Conditions on subset sizes ensuring such polynomials exist

Generalization of prior results to larger subset size regimes

Abstract

For nonempty subsets $S_0, \dots, S_{n-1}$ of a (large enough) finite field $\mathbb{F}$ satisfying $$|S_1|, \dots, |S_{n-1}| > 2 \quad \mathrm{or} \quad |S_1|,|S_{n-1}| > n - 1,$$ we show that there exist $a_0 \in S_0, \dots, a_{n-1} \in S_{n-1}$ such that $$ T^n + a_{n-1}T^{n-1} + \dots + a_1T + a_0 \in \mathbb{F}[T] $$ is a squarefree polynomial.