Jacobian conjecture as a problem on integral points on affine curves

TL;DR

This paper links the Jacobian conjecture over algebraic number fields to the existence of integral points on affine curves, suggesting that a counterexample would imply the absence of integer solutions on certain affine curves.

Contribution

It reformulates the Jacobian conjecture as an existence problem for integral points on affine curves, providing a new perspective on this longstanding problem.

Findings

If the Jacobian conjecture over \\mathbb{C} is false, then specific polynomial maps lack non-zero integer solutions.

Counterexamples to the conjecture correspond to affine curves with no non-zero integer points.

The paper characterizes potential counterexamples in terms of polynomial maps with particular algebraic structures.

Abstract

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then for some $n\gg 1$ there exists a counterexample $F\in \mathbb{Z}[X]^n$ of the form $F_i(X)=X_i+ (a_{i1}X_1+\dots+a_{in}X_n)^{d_i}$, $a_{ij}\in \Z$, $d_i=2;3 $, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\dots=F_n(X)$ has no non-zero integer points.