On Projections of Semi-algebraic Sets Defined by Few Quadratic Inequalities

Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $Let$\pi$denote the standard projection from$\R^{k + m}$onto$\R^m$. We prove that for any$q >0$, the sum of the first$q$Betti numbers of$\pi(S)$is bounded by$(k + m)^{O(q\ell)}.$We also present an algorithm for computing the the first$q$Betti numbers of$\pi(S)$, whose complexity is$ (k+m)^{2^{O(q\ell)}}.$For fixed$q$and$\ell$, both the bounds are polynomial in$k+m$.