An asymptotically tight bound on the number of connected components of realizable sign conditions

In this paper we prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of connected components of the realizations of all realizable sign conditions of a family of s polynomials in R[X1, . . . , Xk] whose degrees are at most d, is bounded by (2d)k k! s + O(sk−1). This improves the best upper bound known previously, which was 1 2 (8d)k k! s + O(sk−1). The new bound matches asymptotically the lower bound obtained for families of polynomials each of which is a product of generic polynomials of degree one.