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Rationality problem of three-dimensional purely monomial group actions: the last case

Authors: Akinari Hoshi and Yūichi Rikuna
Journal: Math. Comp. 77 (2008), 1823-1829
MSC (2000): Primary 14E08, 12F12; Secondary 13A50, 14E07, 20C10
Published electronically: January 28, 2008
MathSciNet review: 2398796
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Abstract: A $k$-automorphism $\sigma$ of the rational function field $k(x_1,\dots ,x_n)$ is called purely monomial if $\sigma$ sends every variable $x_i$ to a monic Laurent monomial in the variables $x_1,\dots ,x_n$. Let $G$ be a finite subgroup of purely monomial $k$-automorphisms of $k(x_1,\dots ,x_n)$. The rationality problem of the $G$-action is the problem of whether the $G$-fixed field ${{k}\!\left ({{x_1},\dots ,{x_n}}\right )^{G}}$ is $k$-rational, i.e., purely transcendental over $k$, or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

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Additional Information

Akinari Hoshi
Affiliation: Department of Mathematics, School of Education, Waseda University, 1–6–1 Nishi-Waseda Shinjuku-ku, Tokyo 169–8050, Japan
MR Author ID: 714371

Yūichi Rikuna
Affiliation: Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3–4–1 Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan

Received by editor(s): December 8, 2006
Received by editor(s) in revised form: April 30, 2007
Published electronically: January 28, 2008
Article copyright: © Copyright 2008 American Mathematical Society