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 $k+m$. 
Saugata Basu and Thierry Zell