Factorization and Resultants of Partial Differential Operators

Abstract

Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.Differential algebra in several variables has not yet reached the same stage of maturity as algebraic geometry [19]. We focus our effort in creating an algebraic theory of commuting PDOs. 1 There are several standing problems to consider.Problem I: Finding examples. Few and far between until the 1980s, classes of examples have now been developed by at least three methods: (1) representation theory, within the universal enveloping algebra of a Lie algebra, for instance by writing Casimirs; (2) quantization [13,14]; (3) flows of integrable hierarchies, when the flows are interpreted in differential-algebra terms under the boson-fermion correspondence [27].These three methods are related: examples of type (1) can also be given as quantized completely integrable systems as in (2), and the original motivation was that of deforming as in (3) Schrödinger operators in several variables;(3) has the interesting feature that the systematic procedures given (on a Sato-like Grassmannian) produce matrices [27], or n-tuples of commuting operators in n variables [28], as opposed to scalars. We display in Sect. 1 the most advanced examples we know of, whose spectral varieties could be studied by our methods. 1 We use throughout the paper the abbreviations ODO (PDO, DO) for ordinary (partial, (formal) pseudo-)differential operator, gcd for greatest common divisor.