$C^m$ Semialgebraic Sections Over the Plane

Abstract

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C_{loc}^m\left({\mathbb{R}}^2,{\mathbb{R}}^D\right).$ If $\mathcal{H}$ has a section, then it has a semialgebraic section.

51 pages. comments welcome