Multivariable polynomial injections on rational numbers

TL;DR

This paper explores the connection between the Bombieri-Lang conjecture and polynomial injections over rational numbers, suggesting that certain conjectures imply the existence of injective polynomial functions.

Contribution

It establishes a link between deep conjectures in number theory and the existence of polynomial injections over rational fields.

Findings

Conjecture implies existence of polynomial injections over k.

Provides theoretical framework connecting number theory and polynomial mappings.

Suggests new directions for research in polynomial injectivity and rational points.

Abstract

For each number field k, the Bombieri-Lang conjecture for k-rational points on surfaces of general type implies the existence of a polynomial f(x,y) in k[x,y] inducing an injection k x k --> k.