Squarefree polynomials with prescribed coefficients

Abstract

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