S-unit points on analytic hypersurfaces

In analogy with algebraic equations with S-units, we shall deal with S-unit points in an analytic hypersurface, or more generally with values of analytic functions at S-unit points.After proving a general theorem, we shall give diophantine applications to certain problems of integral points on subvarieties of . Also, we shall prove an analogue of a theorem of Masser, important in Mahler's method for transcendence.In the course of the proofs we shall also develop a theory for those algebraic subgroups of whose Zariski closure in An contains the origin. Among others, we shall prove a structure theorem for the family of such subgroups contained in a given analytic hypersurface, obtaining conclusions similar to the case of algebraic varieties.