Some Remarks on the Jacobian Conjecture and Connections with Hilbert's Irreducibility Theorem

We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map, defined over any extension of Q, then P has a polynomial inverse if and only if the range of P contains the cartesian product of n universal Hilbert sets. In our second result, we show that if P is a 2-dimensional Keller map, defined over any algebraic number field, then P is invertible on a set that contains almost all rational integers of K.