Growth of solutions of linear differential equations at a logarithmic singularity

Abstract:

We consider differential equations $Y’ = AY$ with a regular singular point at the origin, where $A$ is an $n \times n$ matrix whose entries are $p$-adic meromorphic functions. If the solution matrix at the origin is of the form $Y = P\exp (\theta \log x)$, where $P$ is an $n \times n$ matrix of meromorphic functions and $\theta$ is an $n \times n$ constant matrix whose Jordan normal form consists of a single block, then we prove that the entries of $P$ have logarithmic growth of order $n - 1$.