Integral points on plane curves and Plane Jacobian Conjecture over number fields

TL;DR

This paper explores the relationship between integral points on plane algebraic curves over number fields and the Plane Jacobian Conjecture, linking classical Diophantine problems with polynomial invertibility.

Contribution

It establishes an equivalence between the Plane Jacobian Conjecture over number fields and a Siegel-type statement about integral points on certain algebraic curves.

Findings

The Plane Jacobian Conjecture over number fields is equivalent to a statement about integral points on curves.

A new connection between polynomial invertibility and Diophantine geometry is demonstrated.

The work extends classical results on integral points to the context of the Jacobian conjecture.

Abstract

Let $K$ be a number field and $O_K$ the ring of integers of $K$. In the spirit of Siegel's theorem on integral points on affine algebraic curves, the plane Jacobian conjecture over $K$ is equivalent to the following statement: if $P,Q\in O_K[x,y]$ and $P_xQ_y-P_yQ_x\equiv 1$, then the curve $P=0$ has more than one integral point.