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

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

2009-02-02

arxiv.org/abs/0902.0210v2

J. Alg. 324 (2010), no. 2, 231--247

Mathematics

Complex Variables

Besides some minor changes and presentation improvements, a counter-example to the image conjecture over char. p>0 has been ad

Scientific paper

10.1016/j.jalgebra.2010.04.022.

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.

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