The convergence Newton polygon of a $p$-adic differential equation I : Affinoid domains of the Berkovich affine line

TL;DR

This paper demonstrates that the radii of convergence for solutions to $p$-adic differential equations on affinoid domains in the Berkovich affine line are continuous and can be described by finite graph structures, revealing their super-harmonic nature.

Contribution

It establishes the continuity and super-harmonicity of convergence radii functions, showing they factor through a finite graph, thus controlling their behavior with finite data.

Findings

Convergence radii are continuous functions on affinoid domains.

These functions factor through a finite graph via retraction.

The radii exhibit super-harmonicity properties.

Abstract

We prove that the radii of convergence of the solutions of a $p$-adic differential equation $\mathcal{F}$ over an affinoid domain $X$ of the Berkovich affine line are continuous functions on $X$ that factorize through the retraction of $X\to\Gamma$ of $X$ onto a finite graph $\Gamma\subseteq X$. We also prove their super-harmonicity properties. Roughly speaking, this finiteness result means that the behavior of the radii as functions on $X$ is controlled by a finite family of data.