Commutative partial differential operators

In one variable, it is possible to describe explicitly the differential operators that commute with a given one, at least when the centralizer of the given operator has rank 1. So far, a generalization of the theory to several variables has been developed (inexplicitly) only for matrices, whose size increases with the number of variables. We propose to develop an algebraic theory of commuting partial differential operators (PDOs) by formulating a generalization of the one-variable techniques, in particular Darboux transformations and differential resultants. In this paper, we present a counter example to a one-variable feature of maximal-commutative rings and some facts, examples and questions on differential resultants.