Uniform existential interpretation of arithmetic in rings of functions of positive characteristic

We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation “\(y=x^{p^{s}}\) or \(x=y^{p^{s}}\) for some integer s≥0”. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over \(\mathbb {Z}[z]\) has solutions in all but finitely many polynomial rings \(\mathbb {F}_{p}[z]\). Analogous consequences are deduced for the rational function fields \(\mathbb {F}_{p}(z)\), over languages with a predicate for the valuation ring at zero.