On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials
Saugata BasuPurdue University, West Lafayette, United States
Sal BaronePurdue University, West Lafayette, USA
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 .
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. 1527–1554DOI 10.4171/JEMS/468