On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

Abstract

We consider a differential system $x\frac{d}{dx}Y=GY$, where $G$ is a $m\times m$ matrix whose coefficients are power series which converge and are bounded on the open unit disc $D(0,1^{-})$. Assume that $G(0)$ is a diagonal matrix with $p$-adic integer coefficients. Then there exists a solution matrix of the form $Y=F\exp (G(0)\log x)$ at $x=0$ if all differences of exponents of the system are $p$-adically non-Liouville numbers. We give an example where $F$ is analytic on the $p$-adic open unit disc and has log-growth greater than $m$. Under some conditions, we prove that if a solution matrix at a generic point has log-growth $\delta$, then $F$ has log-growth $\delta$.