On the characterization of minimal value set polynomials

TL;DR

This paper characterizes all minimal value set polynomials over finite fields with value sets as subfields, providing explicit descriptions, counting formulas, and constructions for new examples.

Contribution

It offers an explicit characterization and enumeration of minimal value set polynomials with subfield value sets, including new constructions and bounds.

Findings

The set of such polynomials forms an $_{q'}$-vector space of a specific dimension.

The paper provides exact counts of these polynomials.

It introduces methods to construct new minimal value set polynomials.

Abstract

We give an explicit characterization of all minimal value set polynomials in $\F_q[x]$ whose set of values is a subfield $\F_{q'}$ of $\F_{q}$. We show that the set of such polynomials, together with the constants of $\F_{q'}$, is an $\F_{q'}$-vector space of dimension $2^{[\F_{q}:\F_{q'}]}$. Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.