Un résultat de connexité pour les variétés analytiques p-adiques: privilège et noethérianité

Abstract

AbstractLet k be a non-Archimedean field, let X be a k-affinoid space and let f1,…,fn, with $n\in \mathbb {N}^*$, be analytic functions over X. If X is irreducible, we prove that the analytic domain $\bigcup _{1\le j\le n} \{x\in X\mid |f_{j}(x)|\ge \varepsilon _{j}\}$ is still irreducible, provided that $(\varepsilon _{1},\ldots ,\varepsilon _{n}) \in \mathbb {R}_{+}^n$ is small enough. Then, for a general X, we precisely describe how the geometric connected components of the spaces $\{x\in X\mid |f(x)|\ge \varepsilon \}$ behave with regards to ε. Finally, we obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions.