An application of Newton–Puiseux charts to the Jacobian problem

Abstract

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps $(f,g)$ whose topological degree is $\le 5$, for maps with $gcd(degf,degg)\le 16$ and for maps with. $gcd(degf,degg)$ equal to 2 times a prime.