A slight generalization of Keller's theorem

TL;DR

This paper proposes a slight generalization of Keller's theorem, showing that certain algebraic conditions involving invertible Jacobians and elements in the function field imply the invertibility of polynomial morphisms.

Contribution

It introduces a new criterion involving powers of linear forms in the function field that guarantees invertibility, extending Keller's original result.

Findings

The generalized condition ensures invertibility of polynomial maps with invertible Jacobian.

The result applies to polynomial rings in multiple variables.

Provides a broader class of invertible polynomial morphisms.

Abstract

The famous Jacobian problem asks: Is a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, invertible? If we add the assumption that $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$, then $f$ is invertible; this result is due to O. H. Keller (1939). We suggest the following slight generalization of Keller's theorem: If $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ is a morphism having an invertible Jacobian, and if there exist $n \geq 1$, $a \in \mathbb{C}(f(x),f(y))^*$ and $b \in \mathbb{C}(f(x),f(y))$ such that $(ax +b)^n \in \mathbb{C}(f(x),f(y))$, then $f$ is invertible. A similar result holds for $\mathbb{C}[x_1,\ldots,x_m]$.