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A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

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Abstract

In this paper we consider the problem of bounding the Betti numbers, b i (S), of a semi-algebraic set S⊂ℝk defined by polynomial inequalities P 1≥0,…,P s ≥0, where P i ∈ℝ[X 1,…,X k ], s<k, and deg (P i )≤2, for 1≤is. We prove that for 0≤ik−1,

$$\begin{array}{lll}\displaystyle b_{i}(S)&\displaystyle \le&\displaystyle \frac{1}{2}+(k-s)+\frac{1}{2}\cdot \sum_{j=0}^{\mathit{min}\{s+1,k-i\}}2^{j}{{s+1}\choose j}{{k}\choose j-1}\\[18pt]&\displaystyle \le &\displaystyle \frac{3}{2}\cdot\biggl(\frac{6ek}{s}\biggr)^{s}+k.\end{array}$$

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.

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Author information

Correspondence to Saugata Basu.

Additional information

The first author was supported in part by an NSF grant CCF-0634907. The second author was partially supported by NSF grant CCF-0634907 and the European RTNetwork Real Algebraic and Analytic Geometry, Contract No. HPRN-CT-2001-00271.

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Basu, S., Kettner, M. A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities. Discrete Comput Geom 39, 734–746 (2008). https://doi.org/10.1007/s00454-007-9001-6

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Keywords

  • Betti numbers
  • Quadratic inequalities
  • Semi-algebraic sets