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A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables
TL;DR: In this article, a deformation of commutative polynomial algebras in even numbers of variables is introduced and applied to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators.
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Abstract: We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [Z3] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [Ke] is reduced to an open problem on this deformation of polynomial algebras.
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Citations
New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture
Wenhua Zhao
- 26 Jan 2011
TL;DR: In this paper, a new proof for the equivalence of the Jacobian Conjecture with a special case of the vanishing conjecture of (homogeneous) quadratic differential operators with constant coefficients was given.
Images of commuting differential operators of order one with constant leading coefficients
TL;DR: In this paper, the Jacobian conjecture on commuting differential operators of polynomial algebras of order one with constant leading coefficients was studied and the image conjecture on these differential operators was proposed.
References
•Book
Orthogonal Polynomials of Several Variables
Charles F. Dunkl,Yuan Xu +1 more
- 19 Mar 2001
TL;DR: In this article, the authors considered the properties of orthogonal polynomials on the unit sphere, root systems and Coxeter groups, and the Summability of Orthogonal expansions.
The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
TL;DR: In this article, the Jacobian Conjecture and reduction to degree 3 have been studied in the context of linearization and unipotent reduction, and a formal version of the reduction theorem has been proposed.
•Book
Polynomial Automorphisms and the Jacobian Conjecture
Arno van den Essen
- 01 Aug 2000
TL;DR: In this paper, the authors give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed, and discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture.
Book Review: C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables
TL;DR: Results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation, which reports on the recent development on the general theory of hospitalisation in several variables.
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