Generic degrees of real polynomial Keller maps with non-dense image

For a real polynomial map of constant nonzero Jacobian determinant, a real Keller map, with non-dense image, we determine the possible generic degrees.

Such a degree is necessarily even, and by Campbell's theorem different from two, hence at least four; we show conversely that every even integer $\geq4$ occurs, realized in dimension three by an explicit member of the deformation family of the July 2026 counterexample to the Jacobian conjecture.

Hence for every $n\geq3$ the generic degrees of real Keller maps $\mathbb R^{n}\to\mathbb R^{n}$ with non-dense image are exactly the even integers $\geq4$; in dimension two the set is empty if the planar Jacobian conjecture holds, and in dimension one it is empty.

Density is meant throughout in the Euclidean topology.

An explicit degree-four real Keller map with empty real fibres along a slice of the target is due to Gallagher, and families realizing every generic degree $\geq3$ over $\mathbb C$ to Gallagher and to the base manuscript; both precede this note.