On log-growth of solutions of \(p\)-adic differential equations with \(p\)-adic exponents (Q2154574)

Summary: We consider a differential system \(x \frac{d}{dx} Y = G Y\), 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 \(\delta\).