ON THE TWO-DIMENSIONAL JACOBIAN CONJECTURE: MAGNUS’ FORMULA REVISITED, I

Abstract

Let $K$ be an algebraically closed field of characteristic 0. For $f,g\in K[x,y]$, when the Jacobian $(\partial f∕\partial x)(\partial g∕\partial y)-(\partial g∕\partial x)(\partial f∕\partial y)$ is a constant, Magnus’ formula describes the relations between the homogeneous degree pieces $f_i$ and $g_i$. We show a more general version of Magnus’ formula, which could provide a potentially useful tool to prove the Jacobian conjecture.