Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties

Let A be a semi-abelian variety, and X a subvariety of A, both defined over a number field. Assume that X does not contain X₁ + X₂ for any positive-dimensional subvarieties X₁, X₂ of A. Let Γ be a subgroup of A(C) of finite rational rank. We give doubly exponential bounds for the size of (X ∩ Γ)\X(Ǭ). Among the ingredients is a uniform bound, doubly exponential in the data, on finite sets which are quantifier-free definable in differentially closed fields. We also give uniform bounds on X ∩ Γ in the case where X contains no translate of any semi-abelian subvariety of A and Γ is a subgroup of A(C) of finite rational rank which has trivial intersection with A(Ǭ). (Here A is assumed to be defined over a number field, but X need not be.)