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

DOI:
https://doi.org/10.1090/S0025-5718-08-02069-3

Published electronically:
January 28, 2008

MathSciNet review:
2398796

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
hoshi@ruri.waseda.jp

**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

Email:
rikuna@moegi.waseda.jp

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