Multivariable Burchnall–Chaundy theory

Abstract

Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A 118, 557-583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

Baker-Akhiezer Modules on Rational Varieties

“…. , f n are the basis of the form (2), satisfying the identity (6). We find such number k that f k satisfies (6) for c k = 0.…”

“…Those who wish to read more widely in the theory of commuting operators of several variables and BA-modules can turn to [6].…”

A Banach Algebra Version of the Sato Grassmannian and Commutative Rings of Differential Operators

“…For now, we encode an existing algebraic construction into our framework that we expect to develop in a future work. Our approach is based to an extent on [28,29] and also in view of related work of other authors that is surveyed in [33] to which we refer for details.…”

“…More specifically, it is within this setting that we interpret the relevant conjugation principles of [32,35] where Baker functions are employed. Certain generalizations of [32,35] to several variables are studied in papers such as [28,29] (see also the survey article [33] and references therein). Although this latter work is essentially algebraic-geometric, we outline how the analogous conjugation results can be seen within the proposed setting of operator theory.…”

Factorization and Resultants of Partial Differential Operators

“…. , ∂ n ], taking the greatest common divisor of all of these maximal minor determinants is equivalent to calculating the polynomial resultant [23]; while for ordinary differential operators with variable coefficients the definition reproduces the differential resultant used to construct the spectral curve [31]. The definition of the differential resultant of the operators L i given in [7] is, in our terminology, a particular partial μ-shifted differential resultant of the operators L i with all μ i = 0.…”