A higher dimensional Hilbert irreducibility theorem

TL;DR

Under the assumption of the weak Bombieri-Lang conjecture, the paper generalizes Hilbert's irreducibility theorem to certain algebraic varieties and applies this to show the non-existence of polynomial bijections from  to .

Contribution

It extends Hilbert's irreducibility theorem to higher-dimensional geometrically mordellic varieties assuming the Bombieri-Lang conjecture.

Findings

Generalization of Hilbert's irreducibility theorem under Bombieri-Lang conjecture

Proof of non-existence of polynomial bijections  d7  d7 

Conditional results linking conjectures to polynomial mappings

Abstract

Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of Hilbert's irreducibility theorem holds for families of geometrically mordellic varieties (for instance, families of hyperbolic curves). As an application we prove that, assuming Bombieri-Lang, there are no polynomial bijections $\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$.