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

Abstract

Let $K$ be an algebraically closed field of characteristic 0. When the Jacobian $(\partial f/\partial x)(\partial g/\partial y)-(\partial g/\partial x)(\partial f/\partial y)$ is a constant for $f,g\in K[x,y]$, Magnus' formula from [A. Magnus, Volume preserving transformations in several complex variables, Proc. Amer. Math. Soc. 5 (1954), 256--266] describes the relations between the homogeneous degree pieces $f_i$'s and $g_i$'s. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.