Jacobian conjecture as a problem on integral points on affine curves (Q2054690)

The author relates the famous Jacobian conjecture to problems in Diophantine geometry. Let \(K\) be a number field (a finite extension of \(\mathbb{Q}\)) and \(O_{K}\subset K\) the ring of integers in \(K\) (roots in \(K\) of monic polynomials with coefficients in \(\mathbb{Z}\)). Then the Jacobian conjecture over \(K\) is true if and only if for every Keller map \(F=(F_{1},\ldots ,F_{n})\in O_{K}[X_{1},\ldots,X_{n}]^{n},\) \(\operatorname*{Jac}F\equiv 1,\) with \(F(0)=0,\) the system of equations \[F_{1}(X)=\cdots=F_{n}(X) \] has non-zero points in \(O_{K}^{n}.\) The second main theorem is: If the Jacobian conjecture over \(\mathbb{C}\) is not true then for some \(n\) sufficiently large there exists a Keller map \(F\in\mathbb{Z[}X\mathbb{]}^{n}\) of the form \(F_{i}(X)=X_{i}+\langle a_{i},X\rangle^{d_{i}},\) \(a_{i}\in\mathbb{Z}^{n},\) \(d_{i}\in\{2,3\},\) \(i=1,\ldots,n,\) such that the above system has no non-zero solution in \(\mathbb{Z}^{n}.\)