香蕉视频

Title: Extension and division of $\mathcal{C}^m$ semialgebraic functions

I will discuss $\mathcal{C}^m$ Whitney problems where the given data is semialgebraic and the solution is to be semialgebraic; in particular, questions concerning extension to $\mathbb{R}^n$ of $\mathcal{C}^m$ semialgebraic functions defined on a closed subset, and $\mathcal{C}^m$ semialgebraic solutions of systems of linear equations whose coefficients are semialgebraic functions.

Positive answers are known for for $n=2$ (Fefferman-Luli, 2021) and for general $m,\, n$ modulo a certain loss of differentiability (Bierstone-Campesato-Milman, 2021). I will try to describe the methods of both results. It is not yet evident whether positive answers preserving the differentiability class should be expected, in general.

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