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

TL;DR

This paper establishes a singly exponential bound on the number of homotopy types of limits of semi-algebraic sets, linking it to the additive complexity of defining polynomials, and confirms a conjecture by Basu and Vorobjov.

Contribution

It proves a bound on the homotopy types of limits of semi-algebraic sets based on additive complexity, confirming a prior conjecture.

Findings

Number of homotopy types of limits is singly exponentially bounded.

Bound on homotopy types of semi-algebraic sets defined by formulas with bounded additive complexity.

Confirms conjecture by Basu and Vorobjov.

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 by Basu and Vorobjov [On the number of homotopy types of fibres of a definable map, J. Lond. Math. Soc. (2) 2007, 757--776].