 
Summary: On regular and logarithmic solutions of ordinary
linear differential systems ?
S.A. Abramov 1 , M. Bronstein 2 , and D.E. Khmelnov 1
1 Dorodnicyn Comp. Center of the Russ. Acad. of Sciences, Moscow 119991, Russia,
fsabramov,khmelnovg@ccas.ru
2 INRIA  Caf' e, BP 93, 06902Sophia Antipolis Cedex, France
Abstract. We present an approach to construct all the regular solutions
of systems of linear ordinary differential equations using the desingular
ization algorithm of Abramov & Bronstein (2001) as an auxiliary tool.
A similar approach to find all the solutions with entries in C(z)[log z] is
presented as well, together with a new hybrid method for constructing
the denominator of rational and logarithmic solutions.
1 Introduction
Let C be an algebraically closed field of characteristic 0, z be an indeterminate
over C, and
L = Q ae (z)D ae + \Delta \Delta \Delta +Q 1 (z)D +Q 0 (z); (1)
where D = d=dz and Q ae (z); : : : ; Q 0 (z) 2 C[z]. A regular solution of Ly = 0
(or of L) at a given point z 0 2 C, is a solution of the form (z \Gamma z 0 ) – F (z) with
F (z) 2 C((z \Gamma z 0 ))[log(z \Gamma z 0 )], where C((z \Gamma z 0 )) is the field of (formal) Lau
rent series over C. If F (z) has valuation 0, then – is called the exponent of
