A solution of the problem of van den Essen and Shpilrain. II (Q1758523)

Let \(x=(x_1, x_2, \dots, x_n)\) be \(n\) free commutative variables. A polynomial \(p(x)\) is called a coordinate polynomial if it can be extended to a generating set of cardinality \(n\) of the polynomial algebra \(\mathbb C [x]\). \textit{A. van den Essen} and \textit{V. Shpilrain} [J. Pure Appl. Algebra 119, No. 1, 47--52 (1997; Zbl 0899.13009)] raised the following question: If an endomorphism \(\phi\) of \(\mathbb C[x]\) maps every coordinate polynomial to a coordinate polynomial, must \(\phi\) be an automorphism of \(\mathbb C [x]\)? In the paper under review, the author gives a positive answer to the question above. More precisely, the author proves the following stronger result: Let \(D\) be any positive integer and \(\phi\) an endomorphism of \(\mathbb C[x]\) which maps every coordinate polynomial \(p(x)\) of \(\deg p(x)\geq D\) to a coordinate polynomial, then \(\phi\) is an automorphism of \(\mathbb C [x]\). A more geometric version of this result is also proved and discussed. For part I, see [the author, J. Pure Appl. Algebra 137, No. 1, 49--55 (1999; Zbl 0929.13014)].