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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Inverse degree of an affine space triangular automorphism


Author: Shu Kawaguchi
Journal: Proc. Amer. Math. Soc. 141 (2013), 3353-3360
MSC (2010): Primary 08A35, 13B25, 14J50, 14R10
Published electronically: June 17, 2013
MathSciNet review: 3080158
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Abstract: For any $ \mathbb{Q}$-algebra $ R$ and any triangular automorphism $ f: R^N\to R^N$ with Jacobian one on the affine space, we show that $ \deg (f^{-1})$ is bounded from above by a constant depending only on $ N$ and $ \deg (f)$. This is seen as a generalization of a result by Furter on the affine plane. Our proof uses (a version of) Furter's estimate on nilpotency indices and Abhyankar-Gurjar's formal inversion formula. It follows that when the Jacobian of a triangular automorphism $ f: R^N\to R^N$ is not necessarily equal to one, $ \deg (f^{-1})$ is bounded from above by a constant depending only on $ N$, $ \deg (f)$ and $ \deg (1/\operatorname {Jac}(f))$.


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Additional Information

Shu Kawaguchi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email: kawaguch@math.sci.osaka-u.ac.jp, kawaguch@math.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11631-X
Keywords: Automorphism, degree, inverse, polynomial ring, non-reduced ring
Received by editor(s): October 18, 2010
Received by editor(s) in revised form: December 13, 2011
Published electronically: June 17, 2013
Additional Notes: This work is partially supported by KAKENHI 21740018
Communicated by: Harm Derksen
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.



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