Burchnall–Chaundy theory, Ore extensions and σ-differential operators

Abstract

A classical theorem of J. L. Burchnall and T. W. Chaundy shows that two commuting differential operators P and Q give rise, via a differential resultant, to a complex algebraic curve with equation F (x, y) = 0, such that formally inserting P and Q for x and y in F (x, y) , gives identically zero. In addition, the points on this curve have coordinates which are exactly the eigenvalues associated with the operators P and Q (see the Introduction for a more precise statement). In this paper, we prove a generalization of this result using resultants in Ore extensions.