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 Gamma be a subgroup of A(C) of finite rational rank. We give doubly exponential bounds for the size of (X boolean AND Gamma)\X((Q) over bar). 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 boolean AND Gamma in the case where X contains no translate of any semi-abelian subvariety of A and Gamma is a subgroup of A(C) of finite rational rank which has trivial intersection with A((Q) over bar). (Here A is assumed to be defined over a number field, but X need not be).