Approximate factorization of linear partial differential operators

Abstract

We investigate the problem of the approximate factorization of a linear partial differential operator of arbitrary order and in arbitrarily many variables. For such an operator, we define common obstacles and the associated ring of obstacles to factorizations of the operator, which extends some specified factorization of its symbol. We derive some facts about obstacles (for instance, they are stable, they cannot be of certain order) and give an exhaustive enumeration of obstacles for bivariate operators of order two and three (also a short form was found for these huge formulas).We notice that for a second-order strictly hyperbolic operator, obstacles are exactly the well-known invariants of Laplace. For operators of order three a generalization (not straightforward) is possible. It allows us to find a full system of invariants (with respect to operation conjugation) for operators of order three.