Equations in a Free Q-group

An algorithm is constructed that decides if a given finite system of equations over a free Q-group has a solution, and if it does, finds a solution. 0. Introduction Systems of equations over a group have been widely studied (see, for instance, [4],[5],[11]). This is currently one of the main streams of combinatorial group theory. The problem of deciding if a system of equations in a group has a solution is a generalization of the word and conjugacy problems. Makanin [8] and Razborov [11] proved one of the most significant results in this area: the algorithmic solvability of systems of equations in free groups. Rips and Sela [12] solved equations over hyperbolic groups by reducing the problem to free groups. Myasnikov and Remeslennikov proved that the universal theory is decidable over free A-groups, where A is an integral domain of characteristic 0 and Z is a pure subgroup of A. If Z is not a pure subgroup of A then the structure of a free A-group is much more complicated. It turned out (see [3]) that the crucial case is A = Q. Baumslag [1] proved that the word problem is decidable in free Q-groups. In [6] we proved that the conjugacy problem in these groups is decidable. A subring C of the ring Q is said to be recursive if there is an algorithm which decides whether a given rational number belongs to C. Any subring of Q is of the form Qπ, i.e. generated by the set { 1 p |p ∈ π}, where π is a set of primes. It is not difficult to see that the recursive subrings of Q are exactly the rings Qπ for recursive subsets π. If the set π is not recursive then the Diophantine problem over a free Qπ-group F Qπ is undecidable. Indeed, let a ∈ Fπ be an element which Qπ-generates its own centralizer in F Qπ (i.e. if 1 6= a = b, then r is invertible in Qπ); then an equation x p = a has a solution in Fπ if and only if p ∈ π. The main result of this paper is the following. Theorem 1. Let π be a recursive set of primes. Then there exists an algorithm that decides if a given finite system of equations over a free Qπ-group has a solution, and if it does, finds a solution. In particular, the Diophantine problem over a free Q-group F is decidable. Let A be an arbitrary ring of characteristic 0 with a prime subring Z. The additive isolator IsA(Z) = {a ∈ A | ∃n(na ∈ Z)} of Z in A is a subring of A which Received by the editors April 5, 1996. 1991 Mathematics Subject Classification. Primary 20E05, 20F10. The first author was supported by grants from NSERC and FCAR; the second author was supported by the NSF Grant DMS-9103098. c ©1998 American Mathematical Society