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For any $\ell > 0$, we present an algorithm which takes as input asemi-algebraic set, $S$, defined by $P_1 \leq 0,...,P_s \leq 0$, where each$P_i \in \R[X_1,...,X_k]$ has degree $\leq 2,$ and computes the top $\ell$Betti numbers of $S$, $b_{k-1}(S), ..., b_{k-\ell}(S),$ in polynomial time. Thecomplexity of the algorithm, stated more precisely, is $ \sum_{i=0}^{\ell+2} {s\choose i} k^{2^{O(\min(\ell,s))}}. $ For fixed $\ell$, the complexity of thealgorithm can be expressed as $s^{\ell+2} k^{2^{O(\ell)}},$ which is polynomialin the input parameters $s$ and $k$. To our knowledge this is the firstpolynomial time algorithm for computing non-trivial topological invariants ofsemi-algebraic sets in $\R^k$ defined by polynomial inequalities, where thenumber of inequalities is not fixed and the polynomials are allowed to havedegree greater than one. For fixed $s$, we obtain by letting $\ell = k$, analgorithm for computing all the Betti numbers of $S$ whose complexity is$k^{2^{O(s)}}$.Comment: Some more details added in Sections 6 and

Computing the Top Betti Numbers of Semialgebraic Sets Defined by Quadratic Inequalities in Polynomial Time