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

  • Sal Barone

    Purdue University, West Lafayette, USA
  • Saugata Basu

    Purdue University, West Lafayette, United States

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 defined by a quantifier-free first order formula , where the sum of the additive complexities of the polynomials appearing in is at most , is bounded by . This proves a conjecture made in [5].

Cite this article

Sal Barone, Saugata Basu, 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