Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension

TL;DR

This paper investigates conditions under which two-factor factorizations of linear partial differential operators can be refined into three-factor factorizations, extending results to arbitrary order and variables, and also addresses incomplete factorizations.

Contribution

It provides a general solution for refining two-factor factorizations into three-factor factorizations for LPDOs of any order and dimension, and extends to incomplete factorizations.

Findings

Established conditions for factorization refinement

Extended factorization results to arbitrary order and variables

Proved a general theorem for incomplete factorizations

Abstract

Given a right factor and a left factor of a Linear Partial Differential Operator (LPDO), under which conditions we can refine these two-factor factorizations into one three-factor factorization? This problem is solved for LPDOs of arbitrary order and number of variables. A more general result for the incomplete factorizations of LPDOs is proved as well.