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 X 1 + X 2 for any positive-dimensional subvarieties X 1 , X 2 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.)