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.

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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.