Recursively enumerable sets of polynomials over a finite field are diophantine

We construct a Diophantine interpretation of F-q[W, Z] over F-q[ Z]. Using this together with a previous result that every recursively enumerable (r.e.) relation over F-q[Z] is Diophantine over F-q[W, Z], we will prove that every r.e. relation over F-q[Z] is Diophantine over F-q[Z]. We will also look at recursive infinite base fields F, algebraic over F-p. It turns out that the Diophantine relations over F[Z] are exactly the relations which are r.e. for every recursive presentation.