Recursively enumerable sets of polynomials over a finite field

We prove that a relation over Fq[Z] is recursively enumerable if and only if it is Diophantine over Fq[W,Z]. We do this by first constructing a model of N in Fq[Z], where n is represented by Zn. In a second step, we show that it suffices to eliminate a bounded universal quantifier. Then finally, the hardest part of the proof is to show that we can eliminate this quantifier.