Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic

TL;DR

This paper establishes the undecidability of positive existential theories in rings of entire functions of positive characteristic by interpreting the integers uniformly within these structures, extending classical number theory results.

Contribution

It proves the uniform positive existential interpretability of the integers in rings of entire functions of positive characteristic, and demonstrates undecidability results for their theories.

Findings

Negative solution to Hilbert's tenth problem analogue for these rings

Uniform positive existential interpretability of integers in such rings

Rationality of solutions to certain Pell equations

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 $\{0,1,t,+,\cdot,=\}$-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. As a key intermediate step, we prove a rationality result for the solutions of certain Pell equation (which a priori could be transcendental entire functions).