On Diophantine definability and decidability in some infinite totally real extensions of Q. Alexandra Shlapentokh * Department of Mathematics East Carolina University Greenville, NC 27858 shlapentokh@math.ecu.edu January 5, 2001 Abstract Let M be a number field, let W M be a set of its non-archimedean primes. Then let O M,W M = {x ∈ M|ord t x ≥ 0, ∀t ∈ W M }. Let {p 1 ,...,p r } be a finite set of prime numbers. Let F inf be the field generated by all the p j i -th roots of unity as j →∞ and i =1,...,r. Let K inf be the largest totally real subfield of F inf . Then for any ε> 0, there exists a number field M ⊂ K inf , and a set W M of non-archimedean primes of M such that W M has density greater than 1 − ε, and Z has a Diophantine definition over the integral closure of O M,W M in K inf . 1 Introduction The interest in the questions of Diophantine definability and decidability goes back to a question which was posed by Hilbert: given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z? This question, otherwise known as Hilbert’s 10th problem, has been answered negatively in the work of M. Davis, H. Putnam, J. Robinson and Yu. Matijasevich. (See [2] and [3].) Since the time when this result was obtained, similar questions have been raised for other fields and rings. In other words, let R be a recursive ring. Then, given an arbitrary polynomial equation in several variables over R, is there a uniform algorithm to determine whether such an equation has solutions in R? Arguably the two most interesting and difficult problems in the area concern R = Q and R equal to the ring of algebraic integers of an arbitrary number field. One way to resolve the question of Diophantine decidability negatively over a ring of characteristic 0 is to construct a Diophantine definition of Z over such a ring. This notion is defined below. 1.1 Definition. Let R be a ring and let A ⊂ R. Then we say that A has a Diophantine definition over R if there exists a polynomial f (t, x 1 ,...,x n ) ∈ R[t, x 1 ,...,x n ] such that for any t ∈ R, ∃x 1 ,...,x n ∈ R, f (t, x 1 , ..., x n )=0 ⇐⇒ t ∈ A. If the quotient field of R is not algebraically closed, we can allow a Diophantine definition to consist of several polynomials without changing the nature of the relationship. (See [3] for more details.) * The research for this paper has been partially supported by NSA grant MDA904-98-1-0510 1