A note on the plane Jacobian conjecture

TL;DR

This paper proves that polynomial functions with irreducible fibers of the same genus are coordinates, implying no counterexamples to the Jacobian conjecture have such fibers, thus advancing understanding of polynomial invertibility.

Contribution

It establishes a new condition under which polynomial functions are coordinates, providing evidence supporting the Jacobian conjecture in the plane case.

Findings

Polynomial functions with irreducible fibers of the same genus are coordinates.

No counterexamples to the Jacobian conjecture have fibers that are irreducible curves of the same genus.

Supports the Jacobian conjecture by ruling out certain types of potential counterexamples.

Abstract

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.