Siegel's theorem on integral points and the Jacobian conjecture over the rational field

TL;DR

This paper proves that certain polynomial maps over the rationals with a specific Jacobian condition are invertible if their fibers contain infinitely many rational points, extending classical results on integral points.

Contribution

It establishes a new criterion for invertibility of polynomial maps over the rationals based on the distribution of rational points in fibers.

Findings

Polynomial maps with Jacobian 1 are invertible under specific fiber conditions.

Infinite rational points in a fiber imply invertibility of the polynomial map.

Connects integral point theory with the invertibility problem in polynomial mappings.

Abstract

It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with $P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber $P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.