On symmetric polynomials having triangular automorphisms (Q1888661)

The authors solve the functional equation \(f(x,y)=f(y,x)=f(x,-y+ax+b)\), where \(f(x,y)\) is a polynomial or a rational function. Let \(\mathbb{C}\) denote the field of complex numbers, \(U_m(x)\) be the \(m\)th Chebyshev polynomial of the second kind, defined by the formula \(U_m(\cos x)=\sin(m+1)x/\sin x\) and let \(\mathcal{A}\subset \mathbb{C}\) be the set of algebraic numbers \(a\not=2\) such that \(U_{m-1}(a/2)=0\) for some \(m\geq 2\). The main result of the paper under review is the following Theorem. Let \(a,b\in \mathbb{C}\) and let \(G\) be the group of \(\mathbb{C}\)-automorphisms of the field \(\mathbb{C}(x,y)\) generated by \(s\) and \(t\) defined as \[ t(x)=y,\quad t(y)=x \;\text{ and }\;s(x)=x,\quad s(y)=-y+ax+b. \] Then, two cases occur: (I) \; If \(a\not\in \mathcal{A}\), then \(\mathbb{C}(x,y)^G=\mathbb{C}(\psi)\) and \(\mathbb{C}[x,y]^G=\mathbb{C}[\psi]\) where \[ \psi(x,y)=x^2+y^2-axy-bx-by. \] (II) \; If \(a\in \mathcal{A}\), then \(\mathbb{C}(x,y)^G=\mathbb{C}(\phi,\psi)\) and \(\mathbb{C}[x,y]^G=\mathbb{C}[\phi,\psi],\) where \(\phi\) is defined by \[ \phi(x,y)=xy\prod_{i=1}^{n-2}(U_i(\tfrac{a}{2})x-U_{i-1}(\tfrac{a}{2})y+\tfrac{b}{2-a} (1-U_{i+1}(\tfrac{a}{2}) +U_i(\tfrac{a}{2}))), \] where \(\psi\) is as above and \(n=n(a)\) is defined as the minimal integer \(\geq 3\) such that \(U_{n-1}(a/2)=0\), \(U_{n-2}(a/2)=-1\) and \(U_n(a/2)=1\). (The existence of such an integer is proved in Lemma 2 of the paper).