$C^m$ Semialgebraic Sections Over the Plane

TL;DR

This paper proves that for semialgebraic bundles over the plane with certain smoothness, the existence of a section guarantees a semialgebraic section, settling a specific two-dimensional conjecture.

Contribution

It establishes that in two dimensions, semialgebraic bundles with sections necessarily admit semialgebraic sections, confirming a conjecture in this case.

Findings

Existence of semialgebraic sections for certain bundles over the plane

Resolution of a two-dimensional conjecture in semialgebraic geometry

Extension of smooth sections to semialgebraic sections

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 $C^m_{loc}\left( \mathbb{R}^{2},\mathbb{R}^{D}\right) .$ If $\mathcal{H}$ has a section, then it has a semialgebraic section.