Negative solutions to three-dimensional monomial Noether problem

Rationality problem for quasi-monomial actions

We give a short survey of the rationality problem for quasi-monomial actions which includes Noether's problem and the rationality problem for algebraic tori, and report some results on rationality problem in three recent papers Hoshi, Kang and Kitayama [HKKi], Hoshi, Kang and Kunyavskii [HKKu] and Hoshi and Yamasaki [HY].

Rationality problem of three-dimensional monomial group actions

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in $\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of $G$ in $\mathrm{GL}(3,\mathbb{…

负指标 Poincaré 级数的单项式

On the monomials of Poincaré Series of negative index

本文研究负指标 Poincaré 级数的单项式等式，证明对满足一定条件的负指标，权重至少50的单项式不相等，且至少93%的负指标满足该条件。