Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets |
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Authors: | Basu |
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Institution: | (1) School of Mathematics and College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA saugata@math.gatech.edu, US |
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Abstract: |
Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that
many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers.
We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic
hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined
by quadratic inequalities. |
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Keywords: | |
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