Rational Solutions of the Mixed Differential Equation and its Application to Factorization of
Differential Operators
If L is a differential operator, V(L) is the solution
space of L, and r is a differential operator for which
r(V(L)) is a subspace of V(L), then r is a G-endomorphism
of V(L) where G is the differential Galois group.
The set of G-endomorphisms of V(L) forms a ring, called
the eigenring. This eigenring is useful for factorizing a
differential operator L.
The purpose of this paper is to compute the eigenring
efficiently. To do this we will show how one can bound
the valuations of the coefficients of r. The computation
of this bound is a bit technical, because it requires
some things about local differential operators. However,
these bounds can be computed very quickly, it takes only
a small percentage of the total time of the eigenring
computation.
The eigenring code is available in Maple V release 5. This
code contains some additional improvements that are not
given in the paper, in particular it contains an improved
bound for "apparent" singularities. Such singularities
occur quite often (e.g. when you take symmetric products
or LCLM's of operators) and
typically have exponents like 0,1,2,..,n-2,n so a number
of consecutive integers starting with 0, then a gap, and
then some more integer(s).
For the implementation see the file eigenring in
the diffop package (see the file DFactorLCLM for an application of eigenring).
Download this paper.
NEW IMPLEMENTATION: If L1, L2 are two operators, then let Hom(L1,L2) denote
the vector space of all operators r in C(x)[Dx] for which r(V(L1))
is contained in V(L2). In other words, all operators r with rational
function coefficients for which r(y) is a solution of L2 for every solution y of L1.
This code is now available for download: Hom.