Uniform Positive Existential Interpretation of the Integers in Rings of Entire Functions of Positive Characteristic

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.