A sharper estimate on the Betti numbers of sets defined by quadratic inequalities
In this paper we consider the problem of bounding the Betti numbers, bi( S), of a semi-algebraic set S. subset of R(k) defined by polynomial inequalities P(1) >= 0, . . . , P(s) >= 0, where P(i) is an element of R[X(1), . . . , X(k)], s < k, and deg(Pi) <= 2, for 1 <= i <= s. We prove that for 0 <= i <= k - 1, [GRAPHICS] [GRAPHICS] This improves the bound of k(O(s)) proved by Barvinok ( in Math. Z. 225: 231-244, 1997). This improvement is made possible by a new approach, whereby we first bound the Betti numbers of non-singular complete intersections of complex projective varieties defined by generic quadratic forms, and use this bound to obtain bounds in the real semi-algebraic case.
betti numbers quadratic inequalities semi-algebraic sets