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On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials

Sal Barone;  Saugata Basu — 2014

Journal of the European Mathematical Society

We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order formula defining the family. As an important consequence, we derive that the number of distinct homotopy types of semi-algebraic subsets of ℝ k defined by a quantifier-free first order formula Φ , where the sum of the additive complexities of the polynomials appearing...

Semi-monotone sets

Saugata Basu;  Andrei Gabrielov;  Nicolai Vorobjov — 2013

Journal of the European Mathematical Society

A coordinate cone in ℝ n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of ℝ n , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone...

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