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Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by asystem 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 forany $q >0$, 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)}}.$ Forfixed $q$ and $\ell$, both the bounds are polynomial in $k+m$.Comment: 21 pages, 1 figure. Requires diagrams.te

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