The two-dimensional Centralizer Conjecture

TL;DR

This paper extends a known result about polynomial Jacobians from complex numbers to any characteristic zero field, proposing a conjecture about polynomial centralizers and linking it to the Jacobian Conjecture.

Contribution

It generalizes a Jacobian-related result to arbitrary characteristic zero fields and introduces the two-dimensional Centralizer Conjecture over integral domains.

Findings

The result holds over any characteristic zero field, not just complex numbers.

The conjecture is proven to be true if the two-dimensional Jacobian Conjecture is true.

It establishes a connection between the Jacobian Conjecture and the Centralizer Conjecture.

Abstract

A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following: Suppose the Jacobian of $A$ and $B$ is invertible in $\mathbb{C}[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in \mathbb{C}[x,y]$. Then $w \in \mathbb{C}[A]$. We show that in CMW's result it is possible to replace $\mathbb{C}$ by any field of characteristic zero, and we conjecture the following 'two-dimensional Centralizer Conjecture over $D$': Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in D[x,y]$, $D$ is an integral domain of characteristic zero. Then $w \in D[A]$. We show that if the famous two-dimensional Jacobian Conjecture is true, then the two-dimensional Centralizer Conjecture is true.