A Note on Jacobian Problem Over \(\mathbb {Z}\)

Abstract

Motivated by the Jacobian problem, this article is concerned with the density of the image set \(F( \mathbb {Z}^n)\) of polynomial maps \(F\in \mathbb {Z}[X_1,\dots ,X_n]^n\) with \(\det DF\equiv 1\). It is shown that if such a map F is not invertible, its image set \(F( \mathbb {Z}^n)\) must be very thin in the lattice \( \mathbb {Z}^n\): (1) for almost all lines l in \( \mathbb {Z}^n\) the numbers \(\texttt {\#}(F^{-1}(l) \cap \mathbb {Z}^n)\) are uniformly bounded; (2) \(\texttt {\#}\{ z\in F( \mathbb {Z}^n): \vert z_i\vert \le B\} \ll B^{n-1}\) as \(B\rightarrow +\infty \), where the implicit constants depend on F.