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,
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.
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