Jacobian conjecture as a problem on integral points on affine curves

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 $F_i(X)=X_i+(a_{i1}{X}_1+\cdots +a_{in}{X}_n{)}^{d_i}$, $a_{ij}\in \mathbb{Z}$, $d_i=2;3$, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\cdots =F_n(X)$ has no non-zero integer points.

arXiv comment9 paper