An improved estimate for the maximal growth order of solutions of linear differential systems (Q1884696)

Consider a linear system of ordinary differential equations \(x^{r+1}\frac {dy}{dx}=Ay\), where \(y\) is a vector with \(n\geq2\) components, \(r\) a positive integer, called the Poincaré-rank of the system, and \(A\) a square formal power series matrix of dimension \(n\) of the form \(A=\sum_{\nu=0}^{\infty}A_{\nu }x^{\nu}\), \(A_{0}\neq0\), where the \(A_{\nu}\) are constant matrices with entries in \(\mathbb C\). Let also \(A_{0}\) be a nilpotent matrix and the system is not reducible as defined by \textit{J. Moser} [Math. Z. 72, 379--398 (1960; Zbl 0117.04902)]. The paper improves the estimate on the maximal growth order \(\rho\) of a fundamental solution of the system. This result can be useful when designing algorithms for \(\rho\) and related other invariants.