Minimal value set polynomials over fields of size $p^3$

Abstract:

For any prime number $p$, and integer $k\geqslant 1$, let $\mathbb {F}_{p^k}$ be the finite field of $p^k$ elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials $F\in \mathbb {F}_{p^k}[x]$ for which the value set $\{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\}$ has the minimum possible size $\left \lfloor (p^k-1)/\deg F \right \rfloor +1$. For $k\leqslant 2$, the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for $k=3$.