Oberseminar Zahlentheorie und Arithmetische Geometrie

André--Oort for sums of powers in C^n

Pila's proof of André--Oort for C^n shows that the discriminants of the isolated special points on a hypersurface in C^n may be bounded by an ineffective constant that depends only on the degree of the hypersurface and the degree of its field of definition. In particular, the constant is independent of the height of the equation defining the hypersurface. Most of the special cases of André--Oort that are known effectively (due to Kühne, Bilu, Binyamini etc.) do not possess the same uniformity as Pila's ineffective result. In this talk, I will describe some results which are both uniform and effective for the family of hypersurfaces: a_1 x_1^m + ... + a_n x_n^m = b, where a_1, ..., a_n, b are rational and m is a positive integer.