A sharper estimate on the Betti numbers of sets defined by quadratic inequalities

Abstract

In this paper we consider the problem of bounding the Betti numbers, $b_i(S)$, of a semi-algebraic set $S\subset {\mathbb{R}}^k$ defined by polynomial inequalities $P_1\ge 0,...,P_s\ge 0$, where $P_i\in \mathbb{R}[X_1,...,X_k]$ and $°(P_i)\le 2$, for $1\le i\le s$. We prove that for $0\le i\le k-1$, \[ b_i(S) \le{1/2}(\sum_{j=0}^{min\{s,k-i\}}{{s}\choose j}{{k+1}\choose {j}}2^{j}). \] In particular, for $2\le s\le \frac{k}{2}$, we have \[ b_i(S)\le {1/2} 3^{s}{{k+1}\choose {s}} \leq {1/2} (\frac{3e(k+1)}{s})^s. \] This improves the bound of $k^{O(s)}$ proved by Barvinok. 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.

12 pages, 1 figure, corrected typo