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

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose complexity is (k +m) O(q`) . For fixed q and `, both the bounds are polynomial in k +m.