Journal of the Mathematical Society of Japan

On the set of fixed points of a polynomial automorphism

Zbigniew JELONEK and Tomasz LENARCIK

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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.

Article information

J. Math. Soc. Japan, Volume 68, Number 3 (2016), 1025-1031.

First available in Project Euclid: 19 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R20: Group actions on affine varieties [See also 13A50, 14L30]

affine variety group of automorphisms fixed point of a polynomial automorphism


JELONEK, Zbigniew; LENARCIK, Tomasz. On the set of fixed points of a polynomial automorphism. J. Math. Soc. Japan 68 (2016), no. 3, 1025--1031. doi:10.2969/jmsj/06831025.

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