Images of commuting differential operators of order one with constant leading coefficients

By Wenhua Zhao. 2009

49 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/dcdf8673b69a54251a3395d5f35a57e6a2aaa156

Citations in the corpus

• Generalizations of the Image Conjecture and the Mathieu Conjecture (2009, background): “Abstract. We first propose a generalization of the image conjecture [ Z3 ] to the commuting differential operators related with classical orthogonal polynomials.”