Invariant factorization of LPDOs

Introduction

Factorization of linear ordinary differential operators (LODOs) is known to be unique and in general, it finally reduces to the solution of a Riccati equation [http://en.wikipedia.org/wiki/Riccati_equation] , i.e. factorization of LODOs is not a constructive procedure.

On the other hand, factorization of linear partial differential operators (LPDOs) though not even unique, can be performed constructively using Beals-Kartashova factorization procedure (BK-factorization). BK-factorization is an explicit algorithm for absolute factorization of a bivariate LPDO of arbitrary order $n$ into linear factors. The word absolute means that the coefficient field is not fixed from the very beginning and that the only demand on the coefficients is that they be smooth, i.e. they belong to an appropriate differential field.

The procedure proposed here is to find a first order left factor (when possible) in contrast to the use of right factorization, which is common in the papers of last few decades. Of course the existence of a certain right factor of a LPDO isequivalent to the existence of a corresponding left factor of the transpose of that operator (see below), so in principle nothing is lost by considering left factorization. Moreover taking transposes is trivial algebraically, so there is also nothinglost from the point of view of algorithmic computation.

Factorization

Operator of order 2

Operator of order 3

Operator of order $n$

Generalized invariants

Examples

Transpose

Approximate BK-factorization

References

* A. Loewy. Ueber vollstaendig reduzible lineare homogene Differentialgleichungen, Math. Annalen 62 , pp.89-117 (1906)

* R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. Theor.Math.Phys. Vol. 145(2), pp. 1510-1523 (2005), [http://www.springerlink.com/content/yx664142514k0217/]

* E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. Theor.Math.Phys. Vol.147(3), pp. 839-846 (2006), [http://www.springerlink.com/content/lp81238030114354/]

ee also

* Partial derivative
* Invariant (mathematics)
* Invariant theory
* Laplace invariant