Endomorphisms preserving coordinates of polynomial algebras

TL;DR

This paper proves that endomorphisms of polynomial algebras over fields and rings that preserve coordinates have constant Jacobians, revealing structural constraints on such transformations.

Contribution

It establishes that endomorphisms preserving coordinates must have a Jacobian that is a nonzero constant, extending known results to both field and ring settings.

Findings

Jacobian of coordinate-preserving endomorphisms is a nonzero constant

Results apply to both field and ring polynomial algebras

Provides conditions for endomorphisms to be automorphisms

Abstract

It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] 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[x_1,...,x_n] (where R is a polynomial ring in 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.