Open Access
July, 2016 On the set of fixed points of a polynomial automorphism
J. Math. Soc. Japan 68(3): 1025-1031 (July, 2016). DOI: 10.2969/jmsj/06831025


Let $\Bbb K$ be an algebraically closed field of characteristic zero. We say that a polynomial automorphism $f: \Bbb K^n\to \Bbb K^n$ is special if the Jacobian of $f$ is equal to 1. We show that every $(n-1)$-dimensional component $H$ of the set ${\rm Fix}(f)$ of fixed points of a non-trivial special polynomial automorphism $f: \Bbb K^n\to\Bbb K^n$ is uniruled. Moreover, we show that if $f$ is non-special and $H$ is an $(n-1)$-dimensional component of the set ${\rm Fix}(f)$, then $H$ is smooth, irreducible and $H={\rm Fix}(f)$. Moreover, for $\Bbb K = \mathbb{C}$ if $f$ is non-special and ${\rm Jac}(f)$ has an infinite order in $\Bbb C^*$, then the Euler characteristic of $H$ is equal to 1.


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Zbigniew JELONEK. Tomasz LENARCIK. "On the set of fixed points of a polynomial automorphism." J. Math. Soc. Japan 68 (3) 1025 - 1031, July, 2016.


Published: July, 2016
First available in Project Euclid: 19 July 2016

zbMATH: 1349.14191
MathSciNet: MR3523536
Digital Object Identifier: 10.2969/jmsj/06831025

Primary: 14R10 , 14R20

Keywords: affine variety , fixed point of a polynomial automorphism , group of automorphisms

Rights: Copyright © 2016 Mathematical Society of Japan

Vol.68 • No. 3 • July, 2016
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