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

We prove that the number of homotopy types of limits of one-parametersemi-algebraic families of closed and bounded semi-algebraic sets is boundedsingly exponentially in the additive complexity of any quantifier-free firstorder formula defining the family. As an important consequence, we derive thatthe number of homotopy types of semi-algebraic subsets of $\re^k$ defined by aquantifier-free first order formula $\Phi$, where the sum of the additivecomplexities of the polynomials appearing in $\Phi$ is at most $a$, is boundedby $2^{(k a)^{O(1)}}$. This proves a conjecture made by Basu and Vorobjov [Onthe number of homotopy types of fibres of a definable map, J. Lond. Math. Soc.(2) 2007, 757--776].