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

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 polyn