$\mathcal C^m$ 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 $\mathbb R$.