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arXiv · 1411.7109

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

Natalia Garcia-Fritz·Hector Pasten
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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).

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Natalia Garcia-Fritz, Hector Pasten. 2014-11-26. Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic. https://arxiv.org/abs/1411.7109

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