Let R be a real closed field, Q⊂R[Y1,...,Yℓ,X1,...,Xk], with \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq
d, Q \in {\mathcal Q}, #({\mathcal Q})=m, and P⊂R[X1,...,Xk] with \deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal
P})=s. Let S⊂Rℓ+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. The complexity of the algorithm is bounded by (ℓsmd)2O(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 known previously. Moreover, for fixed m and k this
algorithm has polynomial time complexity in the remaining parameters.Comment: 24 pages, 3 figure