Computing the Betti Numbers of Semi-algebraic Sets Defined by Partly Quadratic Systems of Polynomials

Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, X1, . . . , Xk], with degY (Q) ≤ 2, degX(Q) ≤ d,Q ∈ Q,#(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P ) ≤ d, P ∈ P,#(P) = s. Let S ⊂ R`+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar algorithm described in [6]. The complexity of the algorithm is bounded by (`smd)2 O(m+k) . The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities [6]. Moreover, for fixed m and k this algorithm has polynomial time complexity in the remaining parameters.