Growth of solutions of linear differential equations at a logarithmic singularity

Abstract

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