Mild parametrizations of power-subanalytic sets

TL;DR

This paper establishes uniform $C^r$ and mild parametrization theorems for families of bounded, definable sets in o-minimal structures, providing bounds on the number of parametrizations and their regularity.

Contribution

It introduces two uniform parametrization theorems for definable families, with explicit bounds and mild regularity conditions, advancing the understanding of parametrizations in o-minimal geometry.

Findings

Existence of $C^r$-parametrizations with $cr^m$ maps, uniform in the family.

Existence of $C$-mild parametrizations for any $C>1$, uniform in the family.

Results apply to families of sets of dimension at most $m$ in $ ext{R}_{an}^{ ext{R}}$.

Abstract

We obtain two uniform parametrization theorems for families of bounded sets definable in $\mathbb R_{an}^{\mathbb R}$. Let $X = \{X_t \subset (0,1)^n \mid t \in T\}$ be a definable family of sets $X_t$ of dimension at most $m$. Firstly, $X_t$ admits a $C^r$-parametrization consisting of $cr^m$ maps for some positive constant $c = c(X)$, which is uniform in $t$. Secondly, $X_t$ admits a $C$-mild parametrization for any $C>1$, which is also uniform in $t$.