JournalsjemsVol. 16, No. 8pp. 1527–1554

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

  • Saugata Basu

    Purdue University, West Lafayette, United States
  • Sal Barone

    Purdue University, West Lafayette, USA
On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials cover
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Abstract

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 Rk\mathbb{R}^k defined by a quantifier-free first order formula Φ\Phi, where the sum of the additive complexities of the polynomials appearing in Φ\Phi is at most aa, is bounded by 2(k+a)O(1)2^{(k+a)^{O(1)}}. This proves a conjecture made in [5].

Cite this article

Saugata Basu, Sal Barone, On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials. J. Eur. Math. Soc. 16 (2014), no. 8, pp. 1527–1554

DOI 10.4171/JEMS/468