Uniformity in the Mordell-lang Conjecture

The Mordell-Lang conjecture and its variants assert subvarieties of algebraic groups can meet certain subgroups only in a finite set of translates of subgroups. In most cases the number of such translates is unknown and it is not even known whether this number may be bounded by a function of the geometric data. In this note we show that some uniformity follows immediately from the finiteness result. The main technical result behind this note is a theorem of Pillay on the stability of the theory of an algebraically closed field with a predicate for a group of Lang type [4]. My interest in these questions was renewed through my reading of the paper [5] in which a version of uniformity for the Mordell-Lang conjecture for abelian varieties is proved. As the reader will see the main result of this note is an immediate corollary of Pillay’s theorem, but somehow it was not noticed earlier.