A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

In this paper, we introduce the notions of S-phi-polynomial and S-minimal value set polynomial where phi is a polynomial over a finite field F-q and S is a finite subset of an algebraic closure of F-q. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are S-phi-minimal value set polynomials for phi = x(m), whose S-value sets can be explicitly computed in terms of the monomial x(m).