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

Abstract

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) ∈ k[x] \ {0} such that if S possesses a solution in \( \bar{k}{\left( {\left( {x - \alpha } \right)} \right)^m} \) for some \( \alpha \in \bar{k} \) and a component of this solution has a nonzero polar part, then d(α) = 0. In the case where k ⊆ ℂ and S possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at α, the equality d(α) = 0 is also satisfied.