A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

In this paper we consider the problem of bounding the Betti numbers, b _i (S), of a semi-algebraic set S⊂ℝ ^k defined by polynomial inequalities P ₁≥0,…,P _s ≥0, where P _i ∈ℝX ₁,…,X _k ], s<k, and deg (P _i )≤2, for 1≤i≤s. We prove that for 0≤i≤k−1, This improves the bound of k ^O(s) proved by Barvinok (in Math. Z. 225:231–244, 1997). This improvement is made possible by a new approach, whereby we first bound the Betti numbers of non-singular complete intersections of complex projective varieties defined by generic quadratic forms, and use this bound to obtain bounds in the real semi-algebraic case. The first author was supported in part by an NSF grant CCF-0634907. The second author was partially supported by NSF grant CCF-0634907 and the European RTNetwork Real Algebraic and Analytic Geometry, Contract No. HPRN-CT-2001-00271.