Interpretable groups in Mann pairs

Abstract

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.

Algebraically Closed Fields with a Generic Multiplicative Character

“…. , g ′ k are multiplicatively independent over H, they are algebraically independent over C(H) by (1). As…”

“…, g n ) ≠ 0. Since P is chosen arbitrarily, g is algebraically independent over C(H), and so we have (1).…”

“…, k}, let ∑ i∈I χ(s i ) nd = 0 denote the system which consists of ∑ i∈I χ(s i ) = 0 and ∑ i∈I ′ χ(s i ) ≠ 0 for each non-empty proper subset I ′ of I. By Mann's theorem, there are α (1) , . .…”

“…Still working in a fixed model (F, K; χ) of ACFC p , we show in Section 6 that every definable set has another description which is comparable to the fact that every definable set in a model of ACF is a finite union of quasi-affine varieties. A special case of the above description is when X = ⋃ α∈D V α where {V α } α∈D is a definable family of varieties over K such that (1) for some k ∈ N, D is a subset of F k definable in the language of rings, (2) V α and V β are disjoint for distinct α, β ∈ D.…”

“…There are also several results that hold in the above mentioned two structures and ought to have suitable analogues in our structure but we have not proven them yet. These include the study of imaginaries and definable groups; see [4] and [1].…”