Thanases Pheidas

An Analogue of Hilbert's Tenth Problem for p-Adic Entire Functions

Let p be a fixed prime integer, other than 2, Q p the field of p -adic numbers, and Ω p the completion of the algebraic closure of Q p . Let R p be the ring of entire functions in one variable t over Ω p ; that is, R p is the ring of functions f : Ω p → Q p such that f(t) is given by a power series around 0, of infinite radius of convergence: and where ∣ a ∣ p is the p -adic norm of a in Ω p . We prove: Theorem A. The positive existential theory of R p in the language L = {0, 1, t, +, ·} is undecidable . Theorem A gives a negative answer to the analogue of Hilberts tenth problem for R p in the language L . Related results include those of [2] where it is shown that the first-order theory of entire functions on the complex plane is undecidable and the similar result for analytic functions on the open unit disk (this is due to Denef and Gromov, communicated to us by Cherlin and is as of now unpublished). It would be desirable to have a similar result in the language which, instead of the variable t , has a predicate for the transcendental (that is, nonconstant) elements of R p . A related problem is the similar problem for meromorphic functions on the real or p -adic plane or on the unit open or closed disk. These problems seem for the moment rather hard in view of the fact that the analogue of Hilberts Tenth Problem for the field of rational functions over the complex numbers (or any algebraically closed field of characteristic zero) is an open problem.

Model Theory with Applications to Algebra and Analysis: Decision problems in Algebra and analogues of Hilbert's tenth problem

Introduction One of the first tasks undertaken by Model Theory was to produce elimination results, for example methods of eliminating quantifiers in formulas of certain structures. In almost all cases those methods have been effective and thus provide algorithms for examining the truth of possible theorems. On the other hand, Godels Incompleteness Theorem and many subsequent results show that in certain structures, constructive elimination is impossible. The current article is a (very incomplete) effort to survey some results of each kind, with a focus on the decidability of existential theories, and ask some questions at the intersection of Logic and Number Theory. It has been written having in mind a mathematician without prior exposition to Model Theory. Our presentation will consist of four parts. Part A deals with positive (decidability) results for analogues of Hilberts tenth problem for substructures of the integers and for certain local rings. Part B focuses on the ‘parametric problem’ and the relevance of Hilberts tenth problem to conjectures of Lang. Part C deals with the analogue of Hilberts tenth problem for rings of Analytic and Meromorphic functions. Part D is an informal discussion on the chances of proving a negative (or could it be positive?) answer to the analogue of Hilberts tenth problem for the field of rational numbers.