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Acta Mathematica Vietnamica

A Note on Jacobian Problem Over Z

icon-email Nguyen Van Chau

Abstract

Motivated by the Jacobian problem, this article is concerned with the density of the image set F(Zn) of polynomial maps FZ[X1,,Xn]n with detDF1. It is shown that if such a map F is not invertible, its image set F(Zn) must be very thin in the lattice Zn: (1) for almost all lines l in Zn the numbers #(F1(l)Zn) are uniformly bounded; (2) #{zF(Zn):|zi|B}Bn1 as B+, where the implicit constants depend on F.