Résume | Consider an algebraic curve on which none of the affine coordinates$x_1,...,x_n$ vanishes identically. The intersection with a fixedmultiplicative relation $x_1^{a_1}...x_n^{a_n}=1$ is usually a finite set.If the exponents are allowed to vary, then we can easily get sets as largeas we like, but it was observed by Bombieri, Zannier, and the speaker thatthese are severely restricted. For example when the curve is defined overthe field of algebraic numbers, then the intersection points usually haveabsolute Weil height which is uniformly bounded above. Further the points onthe curve for which there are two independent relations usually form afinite set. We describe some recent progress on such problems, mostly forvarieties other than curves. In particular the natural analogues of theabove statements hold for a plane, but only after removing some ``anomalouscurves" from the plane. |