Jacobian Conjecture as a Problem on Integral Points on Affine Curves

Abstract

It is shown that the Jacobian conjecture over 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 ≫ 1 there exists a counterexample $F\in \mathbb {Z}[X]^{n}$ of the form $F_{i}(X)=X_{i}+ (a_{i1}X_{1}+\cdots +a_{in}X_{n})^{d_{i}}$ , $a_{ij}\in \mathbb {Z}$ , di = 2;3, $i,j=1,\dots ,n$ , such that the affine curve F1(X) = F2(X) = ⋯ = Fn(X) has no non-zero integer points.