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

Abstract

Let $(F,G)$ be a Jacobian pair with $d=w\text{-deg}(F)$ and $e=w\text{-deg}(G)$ for some direction $w$. A generalized Magnus' formula approximates $G$ as ${\sum }_{\gamma \ge 0}{c}_{\gamma }{F}^{\frac{e-\gamma }{d}}$ for some complex numbers $c_{\gamma }$. We develop an approach to the two-dimensional Jacobian conjecture, aiming to minimize the use of terms corresponding to $\gamma >0$. As an initial step in this approach, we define and study the inner polynomials of $F$ and $G$. The main result of this paper shows that the northeastern vertex of the Newton polygon of each inner polynomial is located within a specific region. As applications of this result, we introduce several conjectures and prove some of them for special cases.