Article ID Journal Published Year Pages File Type
4586981 Journal of Algebra 2010 17 Pages PDF
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 (Bass et al. (1982) [BCW], , van den Essen (2000) [E], , de Bondt (2009) [Bo], ) (hence also the Dixmier conjecture (Dixmier (1968) [D], )) and the vanishing conjecture (Zhao (2007) [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.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory