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

2014/11/26 by Garcia-Fritz, Natalia, Pasten, Hector
#11U05 #12L05 #30G06 #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1411.7109

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,+,⋅,=\-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).

Related

• Hilbert's Tenth Problem over Function Fields of Positive Characteristic Not Containing the Algebraic Closure of a Finite Field
• The Diophantine problem for rings of exponential polynomials
• On the structure of the ring of integer-valued entire functions
• On existential definitions of C.E. subsets of rings of functions of characteristic 0
• Uniform existential definitions of valuations in function fields in one variable