Structure of a class of polynomial maps with invariant factors (Q1378324)

Let \(\lambda_{n,a}(x)=\sum^n_{i=1}X^2_i-a\prod^n_{i=1}X_i\).\ The authors say that a polynomial map \(\varphi\in(\mathbb{R}[x_1,\dots, x_n])^n\) has \(\lambda_{n,a}\) as invariant factor if \(\lambda_{n,a}\circ\varphi=\lambda_{n,a}\). They proved earlier [\textit{J. Peyrière}, \textit{Z.-Y. Wen} and \textit{Z.-X. Wen}, Enseign. Math., II. Sér. 39, No. 1-2, 153-175 (1993; Zbl 0798.20017)], that the set of such \(\varphi\) for \(n=3\), \(a=1\), is a group generated by four elements. The main result of this paper is that the set of such \(\varphi\) for any \(n\) and \(a\neq 0\) is a group generated by a subgroup isomorphic to \(S_n\) and two particular simple elements. Note that this study is related to the discrete Schrödinger equation with finitely many potentials via the so-called trace maps [see for example \textit{J. Peyrière}, in: Beyond quasicrystals, Eds. F. Axel, D. Gratias, Les Houches 1994, 465-480 (1995; Zbl 0882.20017) and \textit{A. Sütő}, ibid. 481-549 (1995; Zbl 0884.34083)].