On a conjecture of polynomials with prescribed range

TL;DR

The paper demonstrates that for certain finite fields, there exist multisets with specific properties such that any polynomial with that range must have degree exceeding a given bound, disproving a previous conjecture.

Contribution

It provides a counterexample to Conjecture 5.1 in the referenced work, showing the conjecture does not hold over finite fields with characteristic greater than 9.

Findings

Existence of multisets with prescribed properties for certain finite fields

Polynomials with prescribed range have degree greater than a bound in these cases

Disproof of the conjecture over finite fields with p > 9

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 $k:=q-\ell -1 \geq 3$.