Smooth points of \(p\)-adic subanalytic sets
The author proves some results on the singular locus of a subanalytic subset of \(\mathbb{Z}_ p^ m\). These results extend to the \(p\)-adic fields real variables theorems by \textit{M. Tamm}. The main results are the following: (1) the singular locus of a subanalytic set is itself subanalytic. (2) if \(f\) is a subanalytic function with domain contained in \(\mathbb{Z}_ p^ m\), then there exists an integer \(L\) such that for any point \(x_ 0 \in \mathbb{Z}^ m_ p\) in a neighborhood of which \(f\) is defined, \(f\) is strongly differentiable up to order \(L\) at \(x_ 0\) if, and only if \(f\) is analytic around \(x_ 0\).
scientific article; zbMATH DE number 492806
scientific article