Recursively enumerable sets of polynomials over a finite field are Diophantine

Abstract

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