 
Summary: Desingularization of leading matrices of systems of linear ordinary differential
equations with polynomial coefficients
Abramov S. A. (sergeyabramov@mail.ru, Computing Centre of the Russian Academy of
Science, Russia), Khmelnov D. E. (dennis_khmelnov@mail.ru, Computing Centre of the
Russian Academy of Science, Russia)
We consider systems of linear ordinary differential equations containing m unknown
functions of a single variable x. Coefficients of the systems are polynomials over a number
field. Each of the systems consists of m independent equations. The equations are of
arbitrary orders. We propose an algorithm which, given a system S of this type, constructs
a nonzero polynomial d(x) such that if S possesses an analytic solution having a singularity
at then the equality d() = 0 is satisfied.
Linear differential equations (scalar or system) with variable coefficients appear in many
areas of mathematics. Solving systems leads however to specific difficulties which do not
appear in the scalar case. Consider the equation
Pr(x)y(r)
+ Pr1(x)y(r1)
+ · · · + P0(x)y = 0. (1)
First suppose that this is a scalar equation. The coefficients P0(x), P1(x), . . . , Pr(x) are
polynomials, and Pr(x) is not identically zero. If a solution of (1) has a singularity at some
point then Pr() = 0.
