Negligibility of projective linear automorphisms (Q1379081)

Let \(k\) be a field, \(s=(a_{ij})_{1\leq i,j\leq n+1}\in GL_{n+1}(k)\), and \(K=k(x_1/x_{n+1},\dots,x_n/x_{n+1},y_1/y_{n+1},\dots,\) \(y_n/y_{n+1})\) be a field of transcendence degree \(2n\) over \(k\). The diagonal action of \(s\) on \(K\) is defined as \[ s(x_j/x_{n+1})=\sum^{n+1}_{i=1} a_{ij} x_i/\sum^{n+1}_{i=1} a_{i,n+1} x_i, \] \[ s(y_j/y_{n+1})=\sum^{n+1}_{i=1} a_{ij} y_i/\sum^{n+1}_{i=1} a_{i,n+1} y_i. \] The authors show that there exist \(z_1,\dots,z_n\in K\) such that \(K=k(x_1/x_{n+1},\dots,x_n/x_{n+1},z_1,\dots,\) \(z_n)\) and \(s(z_i)=z_i\) for \(i=1,\dots,n\). In particular, if \(k(x_1/x_{n+1},\dots,x_n/x_{n+1})^{\langle s\rangle}\) is pure transcendental over \(k\), then \(K^{\langle s\rangle}\) is also pure transcendental over \(k\).