On the two-dimensional Jacobian conjecture: Magnus' formula revisited, II

TL;DR

This paper revisits the two-dimensional Jacobian conjecture by extending Magnus' formula and proposing new conjectures inspired by cluster algebras, aiming to advance towards a proof of the conjecture.

Contribution

It introduces a sequence of new conjectures, including the remainder vanishing conjecture, enhancing the utility of Magnus' formula for the Jacobian conjecture.

Findings

Proved the generalized Magnus' formula in the first paper

Introduced new conjectures inspired by cluster algebras

Proposed that these conjectures could lead to a proof of the Jacobian conjecture

Abstract

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence of new conjectures including the remainder vanishing conjecture. This makes the generalized Magnus' formula become a useful tool to show the two-dimensional Jacobian conjecture. In the forthcoming paper(s), we plan to prove the remainder vanishing conjecture.