Endomorphisms preserving coordinates of polynomial algebras (Q2854041)

Let \(k\) be a field and let \(R_n=k[x_1,\ldots,x_n]\) be a polynomial ring over \(k\) in \(n\) variables. A polynomial \(f\in R_n\) is a \textit{coordinate} if there exist an automorphism of \(R\) taking \(f\) to \(x_1\). If the automorphism is a composition of linear and elementary (the ones fixing at least \((n-1)\) \(x_i\)'s) automorphisms then the coordinate is called \textit{tame}. If \(f\) has degree \(1\) then it is called a \textit{linear coordinate}.NEWLINENEWLINE\textit{A. van den Essen} and \textit{V. Shpilrain} [J. Pure Appl. Algebra 119, No. 1, 47--52 (1997; Zbl 0899.13009)] that if \(n=2\) then every \(k\)-endomorphism of \(R_n\) taking a coordinate to a coordinate is an automorphism. \textit{Z. Jelonek} [J. Pure Appl. Algebra 137, No. 1, 49--55 (1999; Zbl 0929.13014)] showed this is also the case for arbitrary \(n\) if \(k\) is algebraically closed of characteristic zero. The authors prove the following results:NEWLINENEWLINETheorem 1.1. Assume \(k\) is an infinite field, \(n>0\) and \(A=R_n[x_{n+1},\ldots,x_{n+m}]\). If an \(R_n\)-endomorphism of \(A\) takes \(R_n\)-linear coordinates of \(A\) to \(R_n\)-coordinates of \(A\) then its Jacobian belongs to \(k^*\).NEWLINENEWLINETheorem 1.2. Assume the characteristic of \(k\) is zero. If a \(k\)-endomorphism of \(R_n\) takes every tame coordinate to a coordinate then its Jacobian belongs to \(k^*\).NEWLINENEWLINENote that if we replace \(R_n\) in Theorem 1.1 by a non-algebraically closed field then the statement is generally not true [\textit{A. A. Mikhalev} et al., Algebra Colloq. 4, No. 2, 159--162 (1997; Zbl 0899.13010)], [\textit{S.-J. Gong} and \textit{J.-T. Yu}, Commun. Algebra 36, No. 4, 1354--1364 (2008; Zbl 1144.13005)].