Special cases of the Jacobian conjecture

TL;DR

This paper investigates specific cases of the Jacobian conjecture, demonstrating that certain algebraic conditions imply the invertibility of polynomial maps with invertible Jacobian in characteristic zero.

Contribution

It establishes that three algebraic conditions are equivalent and sufficient for invertibility in particular cases of the Jacobian conjecture.

Findings

Equivalence of conditions for invertibility

Normality implies invertibility in these cases

Flatness and separability conditions ensure invertibility

Abstract

The famous Jacobian conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ having an invertible Jacobian is invertible ($K$ is a characteristic zero field). We show that if one of the following three equivalent conditions is satisfied, then $f$ is invertible: $K[f(x),f(y)][x+y]$ is normal; $K[x,y]$ is flat over $K[f(x),f(y)][x+y]$; $K[f(x),f(y)][x+y]$ is separable over $K[f(x),f(y)]$.