A Proof of the Plane Jacobian Conjecture

This paper has been withdrawn by the author for further investigation.

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.

This paper has been withdrawn while the author verifies the literature.

This article has been withdrown by the author.

This paper has been withdrawn by the author, due to a significant error in section 4.3.1.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author due to an error in the sufficient condition given for the proof of the Tate conjecture for Catanese surfaces.

This article has been withdrawn due to a mistake which is explained in version 2.

This paper has been withdrawn by the authors due to crucial error in the main proof (located in Section 2.4). The authors apologize for any inconveniences.

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.