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

Abstract: 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 semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of $s$ polynomials in $\R[X_1,...,X_k]$ whose degrees are at most $d$ is bounded by [ \frac{(2d)^k}{k!}s^k + O(s^{k-1}). ] This improves the best upper bound known previously which was [ {1/2}\frac{(8d)^k}{k!}s^k + O(s^{k-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.