B\'ezoutians and injectivity of polynomial maps

We prove that an endomorphism $f$ of affine space is injective on rational points if its B\'ezoutian is constant. Similarly, $f$ is injective at a given rational point if its reduced B\'ezoutian is constant. We also show that if the Jacobian determinant of $f$ is invertible, then $f$ is injective at a given rational point if and only if its reduced B\'ezoutian is constant.