Solvability of rank one $p$-adic differential and $q$-difference equations over the Amice ring

TL;DR

This paper establishes a precise criterion for solving rank one $p$-adic differential and $q$-difference equations over the Amice ring, extending classical decomposition results to this setting.

Contribution

It provides a necessary and sufficient condition for solvability and extends Birkoff decomposition to the Amice ring.

Findings

Characterization of solvability conditions

Extension of Birkoff decomposition to Amice ring

Enhanced understanding of $p$-adic differential equations

Abstract

We provide a necessary and sufficient condition for the solvability of a rank one differential (resp. $q$-difference) equation over the Amice's ring. We also extend to that ring a Birkoff decomposition result, originally due to Motzkin.