A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities |
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Authors: | Saugata Basu Michael Kettner |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA |
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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⊂ℝ
k
defined by polynomial inequalities P
1≥0,…,P
s
≥0, where P
i
∈ℝX
1,…,X
k
], s<k, and deg (P
i
)≤2, for 1≤i≤s. We prove that for 0≤i≤k−1, 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.
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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Keywords: | Betti numbers Quadratic inequalities Semi-algebraic sets |
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