A Full System of Invariants for Third-Order Linear Partial Differential Operators

摘要

A full system of invariants for a third-order bivariate hyperbolic linear partial differential operator L is found under the gauge transformation g(x{sub}1, X{sub}2){sup}(-1) Lg(x{sub}1, x{sub}2). That is, all other invariants can be obtained from this full system, and two operators are equivalent with respect to the gauge transformations if and only if their full systems of invariants are equal. To obtain the invariants, we generalize the notion of Laplace invariants from the case of order two to that of arbitrary order. This is done through the notion of common obstacles to factorizations into first-order factors. Explicit formulae for the invariants of a general operator are given in terms of the coefficients of the operator. The majority of the results were obtained using Maple 9.5.