Bounding the Betti numbers and computing the Euler-Poincaré characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials
Résumé
Let R be a real closed field, Q subset of R[Y-1, ... , Y-l , X-1, ... , X-k], with deg(Y).(Q) <= 2, deg(X). (Q) <= d, Q is an element of Q, #(Q) = m, and P subset of R[X-1, ... , X-k] with deg(X). (P) <= d, P is an element of P, #(P) = s, and S subset of Rl+k a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0; P >= 0, P <= 0, P <= P boolean OR Q. We prove that the sum of the Betti numbers of S is bounded by
l(2)(O(s + l + m)ld)(k+2m).
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-Poincare characteristic of such sets, generalizing similar algorithms described in [4, 9]. The complexity of the algorithm is bounded by (lsmd)(O(m(m+k))).