New proofs for the Abhyankar-Gurjar inversion formula and the equivalence of the Jacobian conjecture and the vanishing conjecture

Author:
Wenhua Zhao

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3141-3154

MSC (2010):
Primary 14R15, 32W99, 14R10

DOI:
https://doi.org/10.1090/S0002-9939-2011-10744-5

Published electronically:
January 26, 2011

MathSciNet review:
2811269

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Abstract: We first give a new proof and also a new formulation for the Abhyankar-Gurjar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gurjar formula to give a more straightforward proof for the equivalence of the *Jacobian conjecture* with a special case of the *vanishing conjecture* of (homogeneous) quadratic differential operators with constant coefficients.

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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/S0002-9939-2011-10744-5

Keywords:
The Abhyankar-Gurjar inversion formula,
the Jacobian conjecture,
the vanishing conjecture of quadratic differential operators

Received by editor(s):
July 23, 2009

Received by editor(s) in revised form:
August 17, 2010

Published electronically:
January 26, 2011

Additional Notes:
The author has been partially supported by NSA Grant H98230-10-1-0168

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2011
American Mathematical Society

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