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

Abramov, S.; Khmelnov, D.
September 2012
Journal of Mathematical Sciences;Sep2012, Vol. 185 Issue 3, p347
Academic Journal
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.


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