Combinatorial complexity in O-minimal geometry

Authors:
Saugata Basu
Affiliations:
Georgia Tech, Atlanta, GA
Venue:
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Year:
2007

Citing 3
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Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)

Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)

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Abstract

In this paper we prove tight bounds on the combinatorial and topological complexity of sets dened in terms of n denable sets belonging to some fixed denable family of sets in an o-minimal structure. This generalizes the combinatorial parts of similar bounds known in the case of semi-algebraic and semi-Pfaffian sets, and as a result vastly increases the applicability of results on combinatorial and topological complexity of arrangements studied in discrete and computational geometry. As a sample application, we extend a Ramsey-type theorem due to Alon et al. [3], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.