Abstract. Let S ⊂ R k+m be a compact semi-algebraic set defined by P1 ≥ 0,..., Pℓ ≥ 0, where Pi ∈ R[X1,..., Xk, Y1,..., Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ ℓ. Let π 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 π(S) is bounded by (k + m) O(qℓ). We also present an algorithm for computing the the first q Betti numbers of π(S), whose complexity is (k + m) 2O(qℓ). For fixed q and ℓ, both the bounds are polynomial in k + m. 1
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