The Yomdin-Gromov algebraic lemma revisited

TL;DR

This paper revisits Gromov's proof of the Yomdin-Gromov algebraic lemma, providing a clearer, more direct proof that guarantees cellular parameterizations and simplifies the induction process.

Contribution

It offers a simplified proof of the Yomdin-Gromov algebraic lemma that closely follows Gromov's original ideas and introduces cellular parameterizations to avoid technical complexities.

Findings

Proof closely follows Gromov's original presentation

Guarantees cellular parameterizations of semialgebraic sets

Simplifies the induction process in the proof

Abstract

In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub's entropy conjecture for $C^\infty$ maps. The statement was further refined by Gromov, producing what is now known as the Yomdin-Gromov algebraic lemma. Several complete proofs based on Gromov's sketch have appeared in the literature, but these have been considerably more complicated than Gromov's original presentation due to some technical issues. In this note we give a proof that closely follows Gromov's original presentation. We prove a somewhat stronger statement, where the parameterizing maps are guaranteed to be \emph{cellular}. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.