Minimal Value Set Polynomials

TL;DR

This paper characterizes minimal value set polynomials over finite fields, describes their possible value sets, and proposes a conjecture that links these polynomials to Frobenius nonclassical curves, with partial proofs for specific cases.

Contribution

It introduces a comprehensive classification of value sets of MVSPs, formulates a conjecture for their complete characterization, and connects these polynomials to algebraic curves over finite fields.

Findings

Complete list of MVSPs with affine subspace value sets

Conjecture on all MVSPs with given value sets

Confirmed cases for q=p, p^2, p^3, and partial for q=p^4

Abstract

A well-known problem in the theory of polynomials over finite fields is the characterization of minimal value set polynomials (MVSPs) over the finite field $\mathbb{F}_q$, where $q = p^n$. These are the nonconstant polynomials $F \in \mathbb{F}_q[x]$ whose value set $V_F = \{F(a) : a \in \mathbb{F}_q\}$ has the smallest possible size, namely $\lceil \frac{q}{\deg(F)} \rceil$. In this paper, we describe the family $\mathcal{A}_q$ of all subsets $S \subseteq \mathbb{F}_q$ with $\# S>2$ that can be realized as the value set of an MVSP $F \in \mathbb{F}_q[x]$. Affine subspaces of $\mathbb{F}_q$ are a fundamental type of set in $\mathcal{A}_q$, and we provide the complete list of all MVSPs with such value sets. Building on this, we present a conjecture that characterizes all MVSPs $F \in \mathbb{F}_q[x]$ with $V_F=S$ for any $S \in \mathcal{A}_q$. The conjecture is confirmed by prior results for $q \in\left\{p, p^2, p^3\right\}$ or $\# S \geq p^{n / 2}$, and additional instances, including the cases for $q=p^4$ and $\# S>p^{n / 2-1}$, are proved. We further show that the conjecture leads to the complete characterization of the $\mathbb{F}_q$-Frobenius nonclassical curves of type $y^d=f(x)$, which we establish as a theorem for $q=p^4$.