论文标题
关于交叉比例ii领域的合理性问题
On a rationality problem for fields of cross-ratios II
论文作者
论文摘要
令$ k $为一个字段，$ x_1，\ dots，x_n $为独立变量，$ l_n = k（x_1，\ dots，x_n）$。对称组$σ_n$通过排列变量对$ l_n $起作用，而投影线性组$ \ text {pgl} _2 $由\ [\ begin {pmatrix} a＆b \\\ c＆d en x_i + d} \]对于每个$ i = 1，\ ldots，n $。固定字段$ l_n^{\ text {pgl} _2} $称为“交叉比例的字段”。给定一个子组$ s \subsetσ_n$，H。tsunogai询问$ l_n^s $合理于$ k_n^s $。当$ n \ geqslant 5 $第二作者表明$ l_n^s $在$ k_n^s $上是合理的，并且仅当$ s $具有$ \ {1，\ dots，n \} $中的奇数序列。在本文中，我们回答了tsunogai的问题，以$ n \ leqslant 4 $。
Let $k$ be a field, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $Σ_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon x_i \mapsto \frac{a x_i + b}{c x_i + d} \] for each $i = 1, \ldots, n$. The fixed field $L_n^{\text{PGL}_2}$ is called "the field of cross-ratios". Given a subgroup $S \subset Σ_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n \geqslant 5$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if $S$ has an orbit of odd order in $\{ 1, \dots, n \}$. In this paper we answer Tsunogai's question for $n \leqslant 4$.
