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

Abstract

We determine the possible generic degrees of real polynomial Keller maps with non-dense image, where density is understood in the Euclidean topology. For every dimension $n\geq3$, these degrees are exactly the even integers $d\geq4$. They are realized in dimension three by an explicit family $G_d\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$, and hence in every higher dimension by stabilization. In dimension two, any such degree is an even integer at least six, and no such maps exist if the planar Jacobian conjecture holds; in dimension one, none exist. For the family $G_d$, we describe the image exactly and show that no $G_d$ omits a half-space. We also show that the maximal cardinality of a real fibre is not determined by the generic degree and is not uniformly bounded in this class.