Abstract. In this paper, we study an algebraically closed field Ω expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup Γ. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple (Ω, k, Γ). This enables us to characterize the interpretable groups when Γ is divisible. Every interpretable group H in (Ω, k, Γ) is, up to isogeny, an extension of a direct sum of krational points of an algebraic group defined over k and an interpretable abelian group in Γ by an interpretable group N , which is the quotient of an algebraic group by a subgroup N 1 , which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in Γ.
IntroductionLet Ω be an algebraically closed ambient field, the field k be a proper subfield of Ω and Γ be a multiplicative subgroup of Ω × . We begin by defining a uniform version of the Mann property which was introduced in [5]. First, we recall the Mann property. Consider an equation(1) a 1 x 1 + · · · + a n x n = 1 with n ≥ 1 and a i ∈ k × . A solution (g 1 , ..., g n ) of this equation is called non-degenerate if for every non-empty subset I of {1, 2, ..., n}, the sum i∈I a i g i is not zero. We say that Γ has the Mann property over k if every such equation (1) has only finitely many non-degenerate solutions in Γ. It was proved by L. van den Dries and A. Günaydın [4, Proposition 5.6] that the Mann property is global, which means if we choose a i to be in Ω in (1) then this still gives finitely many non-degenerate solutions in Γ. We say that (k, Γ) is a Mann pair if for all n there is a finite subset Γ(n) of Γ such that for all a 1 , ..., a n in k × all non-degenerate solutions of (1) in Γ lie in Γ(n). This is a uniform version of the Mann property and the finiteness condition does not depend on the given parameters a 1 , ..., a n . In particular, the group Γ has the Mann property if (k, Γ) is a Mann pair. Given a 1 , ..., a n in k × , let N d(a 1 , ..., a n ) be the set of all non-degenerate solutions of the equation (1) lying in Γ. Therefore, the group Γ has the Mann property if and only if for each tuple (a 1 , ..., a n ) the set N d(a 1 , ..., a n ) is finite, and (k, Γ) is a Mann pair if and only if for each n ≥ 1 the whole union (a 1 ,...,an)∈k × N d(a 1 , ..., a n ) is finite.1991 Mathematics Subject Classification. 03C45.
