A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables

TL;DR

This paper introduces a new deformation of commutative polynomial algebras in even variables, exploring its connections to orthogonal polynomials, differential operators, and reformulating the Jacobian conjecture.

Contribution

It presents a novel deformation of polynomial algebras and links it to classical problems like the Jacobian conjecture, providing new perspectives and reformulations.

Findings

Deformation connects to Laguerre orthogonal polynomials

Reformulation of the image conjecture using deformed algebras

Reduction of the Jacobian conjecture to an open problem in the deformation

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.