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

Abstract

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).

Bézout domains and lattice-valued modules

“…Finally let us note that, contrary to the other examples of rings we considered, it is an open question whether the positive existential theory of H(C) in the language of rings expanded with a new constant symbol interpreted by the identity function of H(C) is decidable [11,Problem 1.1].…”

Interpreting Arithmetic in the First-Order Theory of Addition and Coprimality of Polynomial Rings

“…Undecidability of the theory of a polynomial ring over a domain was shown by R. Robinson, as stated before; for even more general rings of constants, some work is currently being undertaken by E. Naziazeno, M. Barone, and N. Caro [13]. If one also allows a symbol for the indeterminate t in the language, there are much stronger positiveexistential undecidability results for polynomial rings over domains by J. Denef [5,6], for rings of entire functions over fields of characteristic zero by L. Lipschitz and T. Pheidas [12] and for rings of entire functions over fields of positive characteristic by N. Garcia-Fritz and H. Pasten [8]. For more results in this direction we direct the reader to the surveys [14,15].…”

The Diophantine problem for rings of exponential polynomials

“…The first order theory of H C over L z is undecidable [Ro51]. If instead of H C one considers the ring of rigid analytic functions in one variable z over a non-archimedean field k, then undecidability of the positive existential theory over L z is proved in [LP95] when k has characteristic 0, and in [GP15] when k has positive characteristic. A negative solution to the analogue for Hilbert's tenth problem for rings of complex holomorphic functions in at least two variables is proved in [PhV18] over a language including the variables and a predicate for evaluation.…”