Multiple Factorizations of Bivariate Linear Partial Differential Operators

TL;DR

This paper investigates multiple factorizations of bivariate linear partial differential operators of orders three and four, establishing conditions for their factorization structures and providing explicit formulas for complete reducibility.

Contribution

It characterizes when third-order LPDOs have multiple factorizations with coprime symbols and provides explicit criteria for complete reducibility of such operators.

Findings

Third-order LPDOs have a specific factorization structure with coprime symbols.

The coprimality condition is essential for certain factorizations.

Explicit formulas for complete reducibility of LPDOs are derived.

Abstract

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also.