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