${\mathcal{C}}^m$ solutions of semialgebraic equations on curves

Abstract

Consider a system of equations $A(x)\cdot F(x)=B(x)$ on a subset $X$ of ${\mathbb{R}}^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a ${\mathcal{C}}^m$ vector-valued function $f$ on ${\mathbb{R}}^n$, and ask whether we can find a ${\mathcal{C}}^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X=1$. The methods are based on an article of Fefferman and Luli for $X\subset {\mathbb{R}}^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X=1$, or in the case of the semialgebraic Whitney extension problem in ${\mathbb{R}}^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb{R}$.