On Dru\.zkowski's morphisms of cubic linear type

We use theorems of Muller-Quade and Steinwandt, Scheja and Storch, and van der Waerden to study Druzkowski's morphisms of cubic linear type with invertible Jacobian. In particular, we compare the degree of such morphisms with the dimensions of various related vector spaces. These comparisons result in an inequality that, if true, would show that morphisms of cubic linear type with invertible Jacobian are injective, finite, and induce an equality of function fields.