Abstract

We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is uniformly positive existentially interpretable in the class of {0, 1, t, +, ., =}-structures consisting of positive characteristic rings of entire functions on the variable t. From this we deduce uniform undecidability results for the positive existential theory of such structures. (C) 2015 Elsevier Inc. All rights reserved.