Minimal value set polynomials over fields of size 𝑝³

Abstract

For any prime number p p , and integer k ⩾ 1 k\geqslant 1 , let F p k \mathbb {F}_{p^k} be the finite field of p k p^k elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ F p k [ x ] F\in \mathbb {F}_{p^k}[x] for which the value set { F ( α ) : α ∈ F p k } \{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\} has the minimum possible size ⌊ ( p k − 1 ) / deg ⁡ F ⌋ + 1 \left \lfloor (p^k-1)/\deg F \right \rfloor +1 . For k ⩽ 2 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 k=3 .