文档介绍：Computer Algebra Algorithms for Linear
Ordinary Differential and Difference equations
Manuel Bronstein
Abstract. Galois theory has now produced algorithms for solving linear ordi-
nary differential and difference equations in closed form. In addition, recent
algorithmic advances have made those algorithms effective and implementable
puter algebra systems. After introducing the relevant parts of the the-
ory, we describe the latest algorithms for solving such equations.
1. Introduction
Linear ordinary differential equations are equations (resp. systems) of the form
n
X diy(x) dY (x)
a (x) = 0 resp. = A(x)Y (x) ,
i dxi dx
i=0
while linear ordinary difference equations are equations (resp. systems) of the form
n
X
ai(x)y(x + i) = 0 (resp. Y (x + 1) = A(x)Y (x)) ,
i=0
where in both cases the unknown(s) and the coefficients are functions of the contin-
uous or discrete variable x. Similarities between those equations have been noticed
and used for a long time, to the point that algebraic algorithms based on the un-
derlying linear operators allow mon parts of differential or difference
equations solvers to be described, and indeed programmed, in the same algebraic
setting (see . [1, 8]). With the recent discovery of a difference Galois theory [30]
with effective algorithms [10], both problems of deciding whether differential or dif-
ference equations have closed–form solutions are now solved. Furthermore, a recent
reformulation of differential Galois theory [22] allows both cases to presented using
the same algebraic framework, and its interpretation using invariants [12, 13, 29]
has led for the first time to effective implementations puter algebra sys-
tems. After describing mon algebraic setting (below) and outlining the
Galois theory required (section 2), we describe putation of invariants of
differential equations (section 3) and of Liouvillian solutions of difference equa-
tions (section 4). All fields in this paper mutative, rings