Sums of Betti numbers in arbitrary characteristic -1 Sums of Betti numbers in arbitrary characteristic

Sums of Betti numbers in arbitrary characteristic Nicholas M. Katz Introduction In [Mil], Milnor gave an explicit upper bound for the sum of the Betti numbers of a complex affine algebraic variety V. If V is defined in ^N, N ≥ 1, by r ≥ 1 equations Fi, i =1 to r, all of degree ≤ d, Milnor showed ‡i h i(V, $) ≤ d(2d-1)2N-1. Oleinik [Ol] and Thom [Th] gave similar results. It is standard (cf. the proof below that Theorem 2 implies Theorem 1) to infer from Milnor's result an explicit upper bound for the sum of the compact Betti numbers: one finds ‡i h i c(V, $) ≤ 2 r(1 + rd)(1 + 2rd)2N+1.