Fortifying the Yomdin-Gromov Algebraic Lemma

TL;DR

This paper generalizes the Yomdin-Gromov Algebraic Lemma by providing sharp cylindrical parametrizations with bounded $C^{r}$ norm in o-minimal structures, introducing forts to encode combinatorial structures.

Contribution

It introduces forts as a new geometric object to encode cylindrical cell decompositions and extends the lemma to the o-minimal setting with sharper bounds.

Findings

Sharp cylindrical parametrizations with bounded $C^{r}$ norm in o-minimal structures.

Forts as a new tool to encode combinatorial structures of decompositions.

Generalization and strengthening of the Yomdin-Gromov Algebraic Lemma.

Abstract

We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts, geometrical objects encoding the combinatorial structure of cylindrical cell decompositions in o-minimal geometry. Cylindical decompositions, refinements of such decompositions, and cylindrical parametrizations of such decomposition become morphisms in the category of forts. We formulate and prove the above results in the language of forts.