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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Multivariable Burchnall–Chaundy theory

Published:22 June 2007https://doi.org/10.1098/rsta.2007.2064

    Abstract

    Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A118, 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.

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