Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials.

Abstract

Let R be a real closed field, , with degY(Q)2, degX(Q)d, , , and with degX(P)d, , . Let SRℓ+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P=0, P0, P0, . We describe an algorithm for computing the Betti numbers of S generalizing a similar algorithm described in [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. The complexity of the algorithm is bounded by ((ℓ+1)(s+1)(m+1)(d+1))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 [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. Moreover, for fixed m and k this algorithm has polynomial time complexity in the remaining parameters.