Abstract

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 or for some integer sa parts per thousand yen0. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over has solutions in all but finitely many polynomial rings . Analogous consequences are deduced for the rational function fields , over languages with a predicate for the valuation ring at zero.