Desingularization of leading matrices of systems of linear ordinary differential equations with polynomial coefficients 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) + Pr-1(x)y(r-1) + · · · + 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.