Submission date: 26 November 2014.

Abstract:

[Research motivated by an AIM meeting in 2013]
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.
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).

Mathematics Subject Classification: 11U05, 30G06, 12L05

Keywords and phrases:

Full text arXiv 1411.7109: pdf, ps.