On the characterization of minimal value set polynomials

We give an explicit characterization of all minimal value set polynomials in Fq[x] whose set of values is a subfield Fq′ of Fq. We show that the set of such polynomials, together with the constants of Fq′, is an Fq′-vector space of dimension 2[Fq:Fq′]. 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.