Multivariable Burchnall–Chaundy theory-Reference-Cited by-同舟云学术

Multivariable Burchnall–Chaundy theory

Published:2007-06-22 Issue:1867 Volume:366 Page:1155-1177
ISSN:1364-503X
Container-title:Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
language:en
Short-container-title:Phil. Trans. R. Soc. A.

Author:

Previato Emma¹²

Affiliation:

1. School of MathematicsIAS, Princeton, NJ 08540, USA

2. Department of Mathematics and Statistics, Boston UniversityBoston, MA 02215-2411, USA

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.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Link

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2007.2064

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