Abstract

We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic is undecidable. We also establish a negative solution to an analogue of Hilbert's tenth problem for such fields of meromorphic functions, for Diophantine equations including vanishing conditions. These undecidability results are proved by showing that the binary relation is positive existentially definable in such fields. We also prove that the abc conjecture implies a solution to the Erdos-Ulam problem on rational distance sets. These two seemingly distant topics are addressed by a study of power values of bivariate polynomials of the form F(X)G(Y).