Sur les composantes connexes d'une famille d'espaces analytiques p-adiques

TL;DR

This paper investigates the connected components of certain subsets of p-adic analytic spaces defined by inequalities, establishing a finite partition of the parameter space with bounds related to the spectral radius.

Contribution

It proves the existence of a finite partition of the positive real line into intervals where the connected components are canonically bijective, with interval bounds in the square root of the spectral radius.

Findings

Finite partition of $ _+$ into intervals with consistent connected components

Interval bounds belong to $ oot{ ho(A)}$

Connected components are canonically in bijection across intervals

Abstract

Let $X=\mathcal{M}(A)$ be an affinoid space and let $f,g \in A$. We study the sets of connected components of the spaces defined by an inequality of the form $|f|\le r|g|$, with $r\ge 0$. We prove that there exists a finite partition of $\mathbb{R}_+$ into intervals where those sets are canonically in bijection and that the bounds of those intervals belong to $\sqrt{\rho(A)}$.