 
Summary: mSparse Solutions of Linear Ordinary Di#erential
Equations with Polynomial Coe#cients
Sergei A. Abramov #
Computer Center of the Russian Academy of Science
Vavilova 40, Moscow 117967, Russia
abramov@ccas.ru, sabramov@cs.msu.su
Abstract
We introduce the notion of msparse power series (e.g. expanding sin x and
cos x at x = 0 gives 2sparse power series: a coe#cient a n of the series can be
nonzero only if remainder(n, 2) is equal to a fixed number). Then we consider the
problem of finding all mpoints of a linear ordinary di#erential equation Ly = 0
with polynomial coe#cients (i.e., the points at which the equation has a solution
in the form of an msparse series). It is easy to find an upper bound for m. We
prove that if m is fixed then either there exists a finite number of mpoints and all
of them can be found or all points are mpoints and L can be factored as L = ”
L#C
where C is an operator of a special kind with constant coe#cients. Additionally we
formulate simple necessary and su#cient conditions for the existence of mpoints
for an irreducible L.
R’esum’e
