On the characterization of minimal value set polynomials

We give an explicit characterization of all minimal value set polynomials in ${\text{F}}_q[x]$ whose set of values is a subfield ${\text{F}}_{q^{\mathrm{\prime }}}$ of ${\text{F}}_q$. We show that the set of such polynomials, together with the constants of ${\text{F}}_{q^{\mathrm{\prime }}}$, is an ${\text{F}}_{q^{\mathrm{\prime }}}$-vector space of dimension $2^{[{\text{F}}_q:{\text{F}}_{q^{\mathrm{\prime }}}]}$. 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.