Hessian nilpotent polynomials and the Jacobian conjecture
Author:
Wenhua Zhao
Journal:
Trans. Amer. Math. Soc. 359 (2007), 249274
MSC (2000):
Primary 33C55, 39B32, 14R15, 31B05
Published electronically:
July 20, 2006
MathSciNet review:
2247890
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Let and let be the Laplace operator. The main goal of the paper is to show that the wellknown 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.
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 A. van den Essen, Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. MR 1790619 (2001j:14082)
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 A. van den Essen and S. Washburn, The Jacobian conjecture for symmetric Jacobian matrices, J. Pure Appl. Algebra 189 (2004), no. 13, 123133. MR 2038568 (2004m:14133)
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 [Ke]
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 [KR]
 V. Kac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, Advanced Series in Mathematical Physics, 2. World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1021978 (90k:17013)
 [M]
 G. Meng, Legendre Transform, Hessian Conjecture and Tree Formula, mathph/0308035.
 [R]
 J. Rauch, Partial Differential Equations, SpringerVerlag, New York, 1991. MR 1223093 (94e:35002)
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 S. Smale, Mathematical Problems for the Next Century, Math. Intelligencer 20, No. 2, 715, 1998. MR 1631413 (99h:01033).
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 M. Takeuchi, Modern spherical functions, Translations of Mathematical Monographs, 135. American Mathematical Society, Providence, RI, 1994. MR 1280269 (96d:22009)
 [Wa]
 S. Wang, A Jacobian criterion for separability, J. Algebra 65 (1980), 453494. 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, 141150, 191. MR 0592226 (82e:14020)
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 W. Zhao, Recurrent Inversion Formulas, math. CV/0305162.
 [Z2]
 W. Zhao, Inversion Problem, Legendre Transform and the Inviscid Burgers' Equation, J. Pure Appl. Algebra 199 (2005), no.13, 299317. MR 2134306 (2006b:14109)
 [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 617904520
Email:
wzhao@ilstu.edu
DOI:
http://dx.doi.org/10.1090/S0002994706038980
PII:
S 00029947(06)038980
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.
