On singular points of solutions of linear differential systems with polynomial coefficients

S. A. Abramov ; D. E. Khmelnov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 1, pp. 3-21

Résumé

We consider systems of linear ordinary differential equations containing $m$ unknown functions of a single variable $x$. The coefficients of the systems are polynomials over a field $k$ of characteristic $0$. Each of the systems consists of $m$ equations independent over $k[x,d/dx]$. The equations are of arbitrary orders. We propose a computer algebra algorithm that, given a system $S$ of this form, constructs a polynomial $d(x)\in k[x]\setminus\{0\}$ such that if $S$ possesses a solution in $\overline k((x-\alpha))^m$ for some $\alpha\in\overline k$ and a component of this solution has a nonzero polar part, then $d(\alpha)=0$. In the case where $k\subseteq\mathbb C$ and $S$ possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at $\alpha$, the equality $d(\alpha)=0$ is also satisfied.