# Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

## TL;DR

The paper establishes a uniform $C^r$-parameterization for definable sets in a power-analytic structure, with bounds depending polynomially on r, and these maps are real analytic.

## Contribution

It provides explicit bounds on the number of real analytic maps needed for $C^r$-parameterizations of definable sets in power-analytic structures.

## Key findings

- Existence of $C^r$-parameterizations with $cr^{m^3}$ maps
- Maps are real analytic and uniform across definable families
- Bounds are explicit and polynomial in r

## Abstract

We show that if $X$ is an $m$-dimensional definable set in $\mathbb{R}^\text{pow}_\text{an}$, the structure of real subanalytic sets with real power maps added, then for any positive integer r there exists a $C^r$-parameterization of X consisting of $cr^{m^3}$ maps for some constant $c$. Moreover, these maps are real analytic and this bound is uniform for a definable family.