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

Abstract

Let K be an algebraically closed field of characteristic 0. When the Jacobian (∂f /∂x)(∂g/∂y) − (∂g/∂x)(∂f /∂y) is a constant for f, g ∈ K[x, y], Magnus' formula from [23] 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.

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

“…One may wonder why we formulated conjecture B the way we did here instead of using known "smaller" trapezoidal shapes (for instance, as given in [9]). The reason is implicitly given in [25], where we associated divisibility conditions with the geometry of Newton polygons. Basically if we consider divisibility by (a power of) a binomial such as x + 1 or x + y, then Lemma 2.1 will be very useful.…”

“…We need the following lemma, whose proof is similar to [25,Lemma 2.6]. (i) There exists a unique polynomial…”

“…We recall the generalized Magnus' formula from [25]. Note that the last condition on c γ implies that every nonzero summand appearing on the right side of (5.1) is in R[F −1/r d ], so must be a rational function.…”

“…[25]). Suppose [F, G] ∈ C. For any direction w = (u, v) ∈ W , let d = w-deg(F + ) and e = w-deg(G + ).…”