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

 

Degree bounds for inverses of polynomial automorphisms


Authors: Charles Ching-an Cheng, Stuart Sui Sheng Wang and Jie Tai Yu
Journal: Proc. Amer. Math. Soc. 120 (1994), 705-707
MSC: Primary 14E07
MathSciNet review: 1195715
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Abstract: It is known that if $ k$ is a field and $ {\mathbf{F}}:k[{X_1}, \ldots ,{X_n}] \to k[{X_1}, \ldots ,{X_n}]$ is a polynomial automorphism, then $ \deg ({{\mathbf{F}}^{ - 1}}) \leqslant {(\deg \,{\mathbf{F}})^{n - 1}}$. We extend this result to the case where $ k$ is a reduced ring. Furthermore, if $ k$ is not a reduced ring, we show that for any integer $ n \geqslant 1$ and any integer $ \lambda \geqslant 0$ there exists a polynomial automorphism $ {\mathbf{F}}$ such that $ \deg ({{\mathbf{F}}^{ - 1}}) = \lambda + {(\deg \,{\mathbf{F}})^{n - 1}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1195715-1
PII: S 0002-9939(1994)1195715-1
Keywords: Automorphism, degree, inverse, polynomial ring, reduced ring, nilpotent element, nilradical
Article copyright: © Copyright 1994 American Mathematical Society