On Log-growth of Solutions of $p$-adic Differential Equations at a Logarithmic Singular Point

Abstract

We consider a differential system $x\frac{d}{dx} Y=GY$ where $G(0)$ is a nilpotent matrix. Then there exists a solution matrix of the form $Y=F \exp(G(0)\log x)$. If a solution matrix at a generic point is of log-growth $\delta$, then we prove that $F$ is of log-growth $\delta$.