On a conjecture of polynomials with prescribed range

Abstract

We show that, for any integer $\ell$ with $q-\sqrt{p} -1 \leq \ell < q-3$ where $q=p^n$ and $p>9$, there exists a multiset $M$ satisfying that $0\in M$ has the highest multiplicity $\ell$ and $\sum_{b\in M} b =0$ such that every polynomial over finite fields $\fq$ with the prescribed range $M$ has degree greater than $\ell$. This implies that Conjecture 5.1. in \cite{gac} is false over finite field $\fq$ for $p > 9$ and $k:=q-\ell -1 \geq 3$.