# 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

## 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 $\mathbb{R}^k$ defined by a quantifier-free first order formula $\Phi$, where the sum of the additive complexities of the polynomials appearing in $\Phi$ is at most $a$, is bounded by $2^{(k+a)^{O(1)}}$. 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