Multivariable Burchnall-Chaundy theory.
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.
Previato
Emma
E
School of Mathematics, IAS, Princeton, NJ 08540, USA. ep@bu.edu