S. Basu, The combinatorial and topological complexity of a single cell, Discrete & Computational Geometry, 29(1) (2003), 41--59. Home/Search Document Details and Download
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Emerging Challenges in Computational Topology - Bern, Eppstein, al. (1999) (2 citations) (Correct)
.... set is also semi algebraic; this was proved only recently [BPR98] What is the combinatorial or topological complexity of such a component Recently, Basu proved tight bounds on the sum of Betti numbers and used it to prove a sharp bound on the combinatorial complexity of a single component [Bas98] However, no efficient 10 algorithm is known for computing a single cell. A related open problem is to develop an efficient stratification scheme for a single component of a semialgebraic set. In some applications even more challenging problems arise. If the obstacles are moving as well as the ....
Saugata Basu. On the combinatorial and topological complexity of a single cell. In Proc. 39th Symp. Foundations of Computer Science (FOCS '98), pages 606--616. IEEE, November 1998.
Different Bounds on the Different Betti Numbers of Semi-Algebraic.. - Basu (2003) Self-citation (Basu) (Correct)
....and thus sets defined in terms of such hypersurfaces can be topologically more complicated in various non intuitive ways. It is often necessary to estimate the topological complexity of arrangements [9] and sometimes these estimates even play a role in bounding the combinatorial complexity, see [4]. An important measure of the topological complexity of a set S is the Betti numbers b i (S) Here and elsewhere in the paper the set S will always be semi algebraic (that is, defined in terms of a finite number of real polynomial equalities and inequalities) and closed and b i (S) will denote ....
....and Bezout s bound on the number of solutions of a system of polynomial equations. This technique does not allow one to prove separate bounds on the individual Betti numbers smaller than the general bound. The first attempt in trying to prove better bounds for higher Betti numbers was made in [4]. It was motivated by the long standing problem in computational geometry of bounding the combinatorial complexity of a single cell in an arrangement of surface patches. The following result was proved in [4] which bounds the higher Betti numbers of a single connected component of a basic ....
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S. Basu, The combinatorial and topological complexity of a single cell, Discrete & Computational Geometry, 29(1) (2003), 41--59.
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