Bounding the Betti numbers and computing the Euler–Poincaré characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials

Abstract

Let R be a real closed field, Q ⊂ R[Y_1 , . . . , Y__l, X_1 , . . . , Xk], with deg_Y(Q) ≤ 2, deg_X(Q) ≤ d, Q ∈ Q, #(Q) = m, and P ⊂ R[X_1, . . . , Xk] with deg_X(P) ≤ d, P ∈ P, #(P) = s, and S ⊂ R__l+k a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We prove that the sum of the Betti numbers of S is bounded by

l_2 (O(s + l + m)ld)k+2_m.

This is a common generalization of previous results in [4] and [3] on bounding the Betti numbers of closed semi-algebraic sets defined by polynomials of degree d and 2, respectively. We also describe an algorithm for computing the Euler–Poincaré characteristic of such sets, e generalizing similar algorithms described in [4, 9]. The complexity of the algorithm is bounded by (lsmd)O(m(m+k)).