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## Some Definability Results in Abstract Kummer Theory

International Mathematical Research Notes. 2014. Vol. 14. P. 3927-4000.
Gavrilovich M., Bays M., Hils M.

Let S be a semiabelian variety over an algebraically closed ﬁeld, and let X be an irreducible subvariety not contained in a translate of a proper algebraic subgroup of S. We show that the number of irreducible components of [n]^−1 (X) is bounded uniformly in n, and moreover that the bound is uniform in families X_t. . We prove this by Galois-theoretic methods. This proof can be formulated purely model theoretically, and applies in the more general context of divisible abelian groups of ﬁnite Morley rank. In this latter context, we deduce a deﬁnability result under the assumption of the deﬁnable multiplicity property (DMP). We give sufﬁcient conditions for ﬁnite Morley rank groups to have the DMP, and hence give examples where our deﬁnability result holds.