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



There is no tame automorphism of $ \mathbb{C}^{3}$ with multidegree $ (3,4,5)$

Author: Marek Karaś
Journal: Proc. Amer. Math. Soc. 139 (2011), 769-775
MSC (2010): Primary 14Rxx, 14R10
Published electronically: November 10, 2010
MathSciNet review: 2745629
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Abstract: Let $ F=(F_{1},\ldots ,F_{n}):\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ be any polynomial mapping. The multidegree of $ F$, denoted $ \textrm{mdeg} F,$ is the sequence of positive integers $ (\deg F_{1},\ldots ,\deg F_{n}).$ In this paper we address the following problem: for which sequence $ (d_{1},\ldots ,d_{n})$ is there an automorphism or a tame automorphism $ F:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ with $ \mdeg F=(d_{1},\ldots ,d_{n})$? We prove, among other things, that there is no tame automorphism $ F:\mathbb{C}^{3}\rightarrow \mathbb{C}^{3}$ with $ \textrm{mdeg} F=(3,4,5)$.

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Marek Karaś
Affiliation: Instytut Matematyki, Uniwersytetu Jagiellońskiego, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Keywords: Polynomial automorphism, tame automorphism, multidegree.
Received by editor(s): February 24, 2009
Published electronically: November 10, 2010
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.