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

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-Kollár 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.

Minor errors corrected. To appear in Advances in Mathematics

References in corpus (1)

• Continuous solutions to algebraic forcing equations

Cited by in corpus (1)

• Nash approximation of differentiable semialgebraic maps