# A deformation of commutative polynomial algebras in even numbers of variables

Open Mathematics (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topWenhua 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

top- [1] Andrews G.F., Askey R., Roy R., Special functions. Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999
- [2] Bass H., Connell E., Wright D., The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 1982, 7, 287–330 http://dx.doi.org/10.1090/S0273-0979-1982-15032-7 Zbl0539.13012
- [3] Björk J.-E., Rings of differential operators, North-Holland Publishing Co., Amsterdam-New York, 1979
- [4] Coutinho S.C., A primer of algebraic D-modules, London Mathematical Society Student Texts, 33, Cambridge University Press, Cambridge, 1995 Zbl0848.16019
- [5] Dunkl C., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, 81, Cambridge University Press, Cambridge, 2001
- [6] van den Essen A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000 Zbl0962.14037
- [7] Filaseta M., Lam T.-Y., On the irreducibility of the generalized Laguerre polynomials, Acta Arith., 2002, 105(2), 177–182 http://dx.doi.org/10.4064/aa105-2-4 Zbl1010.12001
- [8] Keller O.H., Ganze Gremona-Transformationen, Monats. Math. Physik, 1939, 47(1), 299–306 (in German) http://dx.doi.org/10.1007/BF01695502
- [9] Laguerre E., Sur l′ intégrale ${\int}_{0}^{\infty}\frac{{e}^{-x}dx}{x}$ , Bull. Soc. Math. France, 1879, 7, reprinted in Oeuvres, 1971, 1, 428–437 (in French)
- [10] Pólya G., Szegö G., Problems and theorems in analysis, Vol. II,, Revised and enlarged translation by C.E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976 Zbl0359.00003
- [11] Schur I., Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, I, Sitzungsber. Preuss. Akad. Wiss. Berlin Phys.-Math. Kl., 1929, 14, 125–136 (in German) Zbl55.0069.03
- [12] Schur I., Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome, Journal für die reine und angewandte Mathematik, 1931, 165, 52–58 (in German) http://dx.doi.org/10.1515/crll.1931.165.52 Zbl57.0125.05
- [13] Szegö G., Orthogonal Polynomials, 4th edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975
- [14] Wolfram Research, http://functions.wolfram.com/Polynomials/LaguerreL/
- [15] Wolfram Research, http://functions.wolfram.com/Polynomials/LaguerreL3/
- [16] Zhao W., Hessian Nilpotent Polynomials and the Jacobian Conjecture, Trans. Amer. Math. Soc., 2007, 359(1), 249–274 http://dx.doi.org/10.1090/S0002-9947-06-03898-0 Zbl1109.14041
- [17] Zhao W., A Vanishing Conjecture on Differential Operators with Constant Coefficients, Acta Mathematica Vietnamica, 2007, 32(3), 259–285 Zbl1139.14303
- [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] 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] 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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.