Abstract
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.
Citation
Zbigniew JELONEK. Tomasz LENARCIK. "On the set of fixed points of a polynomial automorphism." J. Math. Soc. Japan 68 (3) 1025 - 1031, July, 2016. https://doi.org/10.2969/jmsj/06831025
Information