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).

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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.