Factoring zero-dimensional ideals of linear partial differential operators

Abstract

We present an algorithm for factoring a zero-dimensional left ideal in the ringQ(x, y)[∂x, ∂y], i.e. factoring a linear homogeneous partial differential system whose coefficients belong toQ(x, y), and whose solution space is finite-dimensional overQ. The algorithm computes all the zero-dimensional left ideals containing the given ideal. It generalizes the Beke-Schlesinger algorithm for factoring linear ordinary differential operators, and uses an algorithm for finding hyperexponential solutions of such ideals.

Loewy decomposition of linear differential equations

“…In a series of publications Li, Schwarz and Tsarev [36][37][38] considered such systems of pde's in the plane and showed that a theory similar as for the ordinary case may be developed.…”

Hyperexponential solutions of finite-rank ideals in orthogonal ore rings

“…In recent years there has been work on decomposing these systems into "subsystems" whose solution spaces are of lower dimension. This has been done following either ideal-theoretical [16,17] or module-theoretical [2,21] approaches. In both cases the methods require the computation of hyperexponential solutions of some linear functional systems obtained from either the associated equations or the integrable systems [20].…”

On the Factorization of Differential Modules

“…By factoring the differential modules associated with systems of linear PDE's with finitedimensional solution spaces, the algorithm FactorDiffMod improves the factorization algorithm in [6]. Further work will include the refinement of the step in the algorithm OneDimSubMods which deals with computing hyperexponential solutions of integrable systems and the improvement for computing ranks of parameterized matrices.…”

“…We now apply the algorithm FactorDiffMod to redo Example 1 in [6]. To compute two-dimensional submodules of M , construct the second exterior power ∧ 2 M of M with a basis {f 1 := e 1 ∧ e 2 , f 2 := e 1 ∧ e 3 , f 3 := e 2 ∧ e 3 } and compute the integrable system associated with ∧ 2 M :…”

“…This idea together with some improvements [2,10] on the Beke-Schlesinger algorithm inspired the study of reducing the factorization problem to that of finding hyperexponential solutions of associated systems. Making use of the main algorithm in [5] for computing hyperexponential solutions of systems of linear partial differential equations (PDE's), Li et al [6,7] generalize the Beke-Schlesinger factorization algorithm to systems of linear PDE's (in one or several unknowns) with finite-dimensional solution spaces. However, in their algorithm, there is a combinatorial explosion caused by trying out all the possible sets of leading derivatives for a potential factor.…”