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 ℋ be a semialgebraic bundle with respect to 𝐶^{𝑚}_{𝑙𝑜𝑐}(ℝ², ℝ^𝐷). If ℋ has a section, then it has a semialgebraic section.