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⊂ℝk defined by polynomial inequalities P1≥0,…,Ps≥0, where Pi∈ℝ[X1,…,Xk], s<k, and deg (Pi)≤2, for 1≤i≤s. We prove that for 0≤i≤k−1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll}\displaystyle b_{i}(S)&\displaystyle \le&\displaystyle \frac{1}{2}+(k-s)+\frac{1}{2}\cdot \sum_{j=0}^{\mathit{min}\{s+1,k-i\}}2^{j}{{s+1}\choose j}{{k}\choose j-1}\\[18pt]&\displaystyle \le &\displaystyle \frac{3}{2}\cdot\biggl(\frac{6ek}{s}\biggr)^{s}+k.\end{array}$$\end{document} This improves the bound of kO(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.