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Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 122, Number 3, June 2000
- pp. 439-450
- 10.1353/ajm.2000.0020
- Article
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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.)