A converse to Cartan's Theorem B: The extension property for real analytic and Nash sets (2506.18347v1)

Abstract: In 1957 Cartan proved his celebrated Theorem B and deduced that if $\Omega\subset{\mathbb R}^n$ is an open set and $X$ is a coherent real analytic subset of $\Omega$, then $X$ has the analytic extension property, that is, each real analytic function on $X$ extends to a real analytic function on $\Omega$. The converse implication in its full generality remains unproven. In the literature only special cases of non-coherent real analytic sets $X\subset\Omega$ without the extension property appear, some of them due to Cartan himself: mainly real analytic sets $X\subset\Omega$ that have a visible tail'. We prove the converse implication: If a subset $X$ of $\Omega$ has the analytic extension property, it is a coherent real analytic subset of $\Omega$. The class of sets with the analytic extension property coincides with that of coherent real analytic sets. We extend the previous characterization to the Nash case, which is somehow more demanding, because of its finiteness properties and its disappointing behavior with respect to cohomology of sheaves of Nash function germs. Let $\Omega\subset{\mathbb R}^n$ be an open semialgebraic set and let $X\subset\Omega$ be a set. We prove that each Nash function on $X$ extends to a Nash function on $\Omega$ if and only if $X\subset\Omega$ is a coherent Nash set. Theif' implication goes back to some celebrated results of Coste, Ruiz and Shiota. If $M\subset{\mathbb R}^n$ is a Nash manifold, ${\mathcal C}^\infty$ semialgebraic functions on $M$ coincide with Nash functions on $M$. As an application, we confront the coherence of a Nash set $X\subset\Omega$ with the fact that each ${\mathcal C}^\infty$ semialgebraic function on $X$ is a Nash function on $X$. We provide a full characterization of the semialgebraic sets $S\subset\Omega$ for which ${\mathcal C}^\infty$ semialgebraic functions on $S$ coincide with Nash functions on $S$.