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,+,⋅,=}{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.