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.
Categories: math.NT