(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 3-21

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

S. A. Abramov
D. E. Khmelnov

Abstract

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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 $k\left[x,d/dx\right]$. The equations are of arbitrary orders. We propose a computer algebra algorithm that, given a system $S$ of this form, constructs a polynomial $d\left(x\right)$Î 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 $d\left($a)=0. In the case where $k$Í C and $S$ possesses an analytic solution having a singularity of an arbitrary type (not necessarily a pole) at a, the equality $d\left($a)=0 is also satisfied.

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