A note on non-Robba $p$-adic differential equations

TL;DR

This paper investigates non-Robba $p$-adic differential equations on annuli, proving that solutions are analytic and bounded within specific disks under certain slope conditions of the radius of convergence function.

Contribution

It establishes that solutions to these differential modules are analytic and bounded in the disk determined by the radius of convergence, given a single-slope condition in the logarithmic scale.

Findings

Solutions are analytic and bounded within the radius of convergence.

The radius function has only one slope on the annulus.

Solutions behave well under the single-slope condition.

Abstract

Let $\mathcal{M}$ be a differential module, whose coefficients are analytic elements on an open annulus $I$ ($\subset \bR_{>0}$) in a valued field, complete and algebraically closed of inequal characteristic, and let $R(\mathcal{M}, r)$ be the radius of convergence of its solutions in the neighbourhood of the generic point $t_r$ of absolute value $r$, with $r\in I$. Assume that $R(\mathcal{M}, r)<r$ on $I$ and, in the logarithmic coordinates, the function $r\longrightarrow R(\ mathcal{M}, r)$ has only one slope on $I$. In this paper, we prove that for any $r\in I$, all the solutions of $\mathcal{M}$ in the neighborhood of $t_r$ are analytic and bounded in the disk $D(t_r,R(\mathcal{M},r)^-)$.