Images of Commuting Differential Operators of Order One with Constant Leading Coefficients

TL;DR

This paper investigates properties of commuting differential operators with constant leading coefficients, proposes the image conjecture, and shows its equivalence to major conjectures like the Jacobian and Dixmier conjectures, linking them to Laplace transformations.

Contribution

It introduces the image conjecture for these differential operators and establishes its equivalence to several longstanding conjectures in algebra and analysis.

Findings

Proposes the image conjecture for commuting differential operators.

Shows the equivalence of the image conjecture to the Jacobian, Dixmier, and vanishing conjectures.

Discusses the connection between the image conjecture and multidimensional Laplace transformations.

Abstract

We first study some properties of images of commuting differential operators of polynomial algebras of order one with constant leading coefficients. We then propose what we call the image conjecture on these differential operators and show that the Jacobian conjecture [BCW], [E], [Bo] (hence also the Dixmier conjecture [D]) and the vanishing conjecture [Z3] of differential operators with constant coefficients are actually equivalent to certain special cases of the image conjecture. A connection of the image conjecture, and hence also the Jacobian conjecture, with multidimensional Laplace transformations of polynomials is also discussed.