Endomorphisms Preserving Coordinates of Polynomial Algebras

Abstract

It is proved that the Jacobian of a k-endomorphism of k[x1,…,xn] over a field k of characteristic zero, taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A=:R[x1,…,xn] (where R is a polynomial ring in a finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.