A deformation of commutative polynomial algebras in even numbers of variables

• Autorzy: Wenhua Zhao
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

The generalized Laguerre polynomials; Total symbols of differential operators; The image conjecture; The Jacobian conjecture

Kategorie tematyczne

• 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
• 32W99: None of the above, but in this section
• 14R15: Jacobian problem

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 1
• Strony: 73-97

Daty

wydano

2010-02-01

online

2010-02-02

Twórcy

autor

Wenhua Zhao

• Illinois State University

Bibliografia

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• [6] van den Essen A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000
• [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
• [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
• [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)
• [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
• [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
• [17] Zhao W., A Vanishing Conjecture on Differential Operators with Constant Coefficients, Acta Mathematica Vietnamica, 2007, 32(3), 259–285
• [18] Zhao W., Images of commuting differential operators of order one with constant leading coefficients, preprint available at http://arxiv.org/abs/0902.0210
• [19] Zhao W., Generalizations of the Image Conjecture and the Mathieu Conjecture, J. Pure Appl. Algebra, doi:10.1016/j.jpaa.2009.10.007
• [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

Identyfikatory

DOI

10.2478/s11533-009-0074-8