A deformation of commutative polynomial algebras in even numbers of variables

Wenhua Zhao

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 73-97
  • ISSN: 2391-5455

Abstract

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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 [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.

How to cite

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Wenhua Zhao. "A deformation of commutative polynomial algebras in even numbers of variables." Open Mathematics 8.1 (2010): 73-97. <http://eudml.org/doc/269483>.

@article{WenhuaZhao2010,
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 [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.},
author = {Wenhua Zhao},
journal = {Open Mathematics},
keywords = {The generalized Laguerre polynomials; Total symbols of differential operators; The image conjecture; The Jacobian conjecture; the generalized Laguerre polynomials; total symbols of differential operators; the image conjecture; the Jacobian conjecture},
language = {eng},
number = {1},
pages = {73-97},
title = {A deformation of commutative polynomial algebras in even numbers of variables},
url = {http://eudml.org/doc/269483},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Wenhua Zhao
TI - A deformation of commutative polynomial algebras in even numbers of variables
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 73
EP - 97
AB - 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 [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.
LA - eng
KW - The generalized Laguerre polynomials; Total symbols of differential operators; The image conjecture; The Jacobian conjecture; the generalized Laguerre polynomials; total symbols of differential operators; the image conjecture; the Jacobian conjecture
UR - http://eudml.org/doc/269483
ER -

References

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  17. [17] Zhao W., A Vanishing Conjecture on Differential Operators with Constant Coefficients, Acta Mathematica Vietnamica, 2007, 32(3), 259–285 Zbl1139.14303
  18. [18] Zhao W., Images of commuting differential operators of order one with constant leading coefficients, preprint available at http://arxiv.org/abs/0902.0210 Zbl1197.14064
  19. [19] Zhao W., Generalizations of the Image Conjecture and the Mathieu Conjecture, J. Pure Appl. Algebra, doi:10.1016/j.jpaa.2009.10.007 Zbl1205.33017
  20. [20] Zhao W., New Proofs for the Abhyankar-Gujar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture, preprint available at http://arxiv.org/abs/0907.3991 

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