$\mathcal C^m$ solutions of semialgebraic or definable equations

TL;DR

This paper investigates conditions under which geometric data can be preserved by $ ext{C}^m$ solutions in the Whitney extension and Brenner-Fefferman-Hochster-Kollár problems, focusing on definable and semialgebraic functions with a loss of differentiability.

Contribution

It establishes a relationship between the differentiability of solutions and the data's definability, providing explicit bounds for the differentiability loss in both problems.

Findings

Existence of a function r(m) linking solution smoothness to data smoothness

Solutions can be made definable if data is definable and smoothness exceeds r(m)

Results apply to semialgebraic and o-minimal definable functions

Abstract

We address the question of whether geometric conditions on the given data can be preserved by a solution in (1) the Whitney extension problem, and (2) the Brenner-Fefferman-Hochster-Koll\'ar problem, both for $\mathcal C^m$ functions. Our results involve a certain loss of differentiability. Problem (2) concerns the solution of a system of linear equations $A(x)G(x)=F(x)$, where $A$ is a matrix of functions on $\mathbb R^n$, and $F$, $G$ are vector-valued functions. Suppose the entries of $A(x)$ are semialgebraic (or, more generally, definable in a suitable o-minimal structure). Then we find $r=r(m)$ such that, if $F(x)$ is definable and the system admits a $\mathcal C^r$ solution $G(x)$, then there is a $\mathcal C^m$ definable solution. Likewise in problem (1), given a closed definable subset $X$ of $\mathbb R^n$, we find $r=r(m)$ such that if $g:X\to\mathbb R$ is definable and extends to a $\mathcal C^r$ function on $\mathbb R^n$, then there is a $\mathcal C^m$ definable extension.