On the Betti numbers of real varieties

The object of this note will be to give an upper bound for the sum of the Betti numbers of a real affine algebraic variety. (Added in proof. Similar results have been obtained by R. Thom lO. ) Let F be a variety in the real Cartesian space Rm, defined by polynomial equations /i (xi, ■ ■ ■, xm) = 0, ■ • •, fP (xx, ■ ■ ■, xm) = 0. The qth Betti number of V will mean the rank of the Cech cohomology group Hq (V), using coefficients in some fixed field F. Theorem 2. If each polynomial f, has degree S k, then the sum of the Betti numbers of V is ᵏ (2k — l) m-1. Analogous statements for complex and/or projective varieties will be given at the end. I wish to thank W. May for suggesting this problem to me. Remark A. This is certainly not a best possible estimate. (Compare Remark B. ) In the examples which I know, the sum of the Betti