Hessian nilpotent polynomials and the Jacobian conjecture

Author:
Wenhua Zhao

Journal:
Trans. Amer. Math. Soc. **359** (2007), 249-274

MSC (2000):
Primary 33C55, 39B32, 14R15, 31B05

DOI:
https://doi.org/10.1090/S0002-9947-06-03898-0

Published electronically:
July 20, 2006

MathSciNet review:
2247890

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and let be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the *vanishing conjecture*: for any homogeneous polynomial of degree , if for all , then when , or equivalently, when . It is also shown in this paper that the condition () above is equivalent to the condition that is Hessian nilpotent, i.e. the Hessian matrix is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.

**[Ar]**V. I. Arnol′d,*Mathematical methods of classical mechanics*, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR**0690288****[ABR]**Sheldon Axler, Paul Bourdon, and Wade Ramey,*Harmonic function theory*, 2nd ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 2001. MR**1805196****[BCW]**H. Bass, E. Connell and D. Wright,*The Jacobian conjecture, reduction of degree and formal expansion of the inverse*. Bull. Amer. Math. Soc.**7**(1982), 287-330. MR**0663785 (83k:14028)****[BE1]**Michiel de Bondt and Arno van den Essen,*A reduction of the Jacobian conjecture to the symmetric case*, Proc. Amer. Math. Soc.**133**(2005), no. 8, 2201–2205. MR**2138860**, https://doi.org/10.1090/S0002-9939-05-07570-2**[BE2]**Michiel de Bondt and Arno van den Essen,*Nilpotent symmetric Jacobian matrices and the Jacobian conjecture*, J. Pure Appl. Algebra**193**(2004), no. 1-3, 61–70. MR**2076378**, https://doi.org/10.1016/j.jpaa.2004.03.003**[BE3]**Michiel de Bondt and Arno van den Essen,*Singular Hessians*, J. Algebra**282**(2004), no. 1, 195–204. MR**2095579**, https://doi.org/10.1016/j.jalgebra.2004.08.026**[BE4]**Michiel de Bondt and Arno van den Essen,*Nilpotent symmetric Jacobian matrices and the Jacobian conjecture. II*, J. Pure Appl. Algebra**196**(2005), no. 2-3, 135–148. MR**2110519**, https://doi.org/10.1016/j.jpaa.2004.08.030**[BE5]**M. de Bondt and A. van den Essen,*Hesse and the Jacobian Conjecture*, Report No. 0321, University of Nijmegen, December, 2003.**[E]**Arno van den Essen,*Polynomial automorphisms and the Jacobian conjecture*, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR**1790619****[EW]**Arno van den Essen and Sherwood Washburn,*The Jacobian conjecture for symmetric Jacobian matrices*, J. Pure Appl. Algebra**189**(2004), no. 1-3, 123–133. MR**2038568**, https://doi.org/10.1016/j.jpaa.2003.10.020**[H]**Henryk Iwaniec,*Topics in classical automorphic forms*, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR**1474964****[Ke]**O. H. Keller,*Ganze Gremona-Transformation*, Monats. Math. Physik**47**(1939), 299-306.**[KR]**V. G. Kac and A. K. Raina,*Bombay lectures on highest weight representations of infinite-dimensional Lie algebras*, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR**1021978****[M]**G. Meng,*Legendre Transform, Hessian Conjecture and Tree Formula*, math-ph/0308035.**[R]**Jeffrey Rauch,*Partial differential equations*, Graduate Texts in Mathematics, vol. 128, Springer-Verlag, New York, 1991. MR**1223093****[S]**Steve Smale,*Mathematical problems for the next century*, Math. Intelligencer**20**(1998), no. 2, 7–15. MR**1631413**, https://doi.org/10.1007/BF03025291**[T]**Masaru Takeuchi,*Modern spherical functions*, Translations of Mathematical Monographs, vol. 135, American Mathematical Society, Providence, RI, 1994. Translated from the 1975 Japanese original by Toshinobu Nagura. MR**1280269****[Wa]**S. Wang,*A Jacobian criterion for separability*, J. Algebra**65**(1980), 453-494. MR**0585736 (83e:14010)****[Wr1]**D. Wright,*Ideal Membership Questions Relating to the Jacobian Conjecture.*To appear.**[Wr2]**D. Wright,*The Jacobian Conjecture: Ideal Membership Questions and Recent Advances*, To appear.**[Wr3]**D. Wright,*Personal communications.***[Y]**A. V. Jagzev,*On a problem of O.-H. Keller.*(Russian) Sibirsk. Mat. Zh.**21**(1980), no. 5, 141-150, 191. MR**0592226 (82e:14020)****[Z1]**W. Zhao,*Recurrent Inversion Formulas*, math. CV/0305162.**[Z2]**Wenhua Zhao,*Inversion problem, Legendre transform and inviscid Burgers’ equations*, J. Pure Appl. Algebra**199**(2005), no. 1-3, 299–317. MR**2134306**, https://doi.org/10.1016/j.jpaa.2004.12.026**[Z3]**W. Zhao,*Some Properties and Open Problems of Hessian Nilpotent polynomials*, In preparation.

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Additional Information

**Wenhua Zhao**

Affiliation:
Department of Mathematics, Illinois State University, Normal, Illinois 61790-4520

Email:
wzhao@ilstu.edu

DOI:
https://doi.org/10.1090/S0002-9947-06-03898-0

Keywords:
Hessian nilpotent polynomials,
deformed inversion pairs,
the heat equation,
harmonic polynomials,
the Jacobian conjecture.

Received by editor(s):
October 15, 2004

Received by editor(s) in revised form:
October 26, 2004

Published electronically:
July 20, 2006

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.