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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...

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