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

2026/07/23 by Piotr Migus
Mathematics · #math.AG

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arxiv created 2026/07/30 · arxiv updated 2026/07/31

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≥3, these degrees are exactly the even integers d≥4. They are realized in dimension three by an explicit family G_d\colonℝ³→ℝ³, 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. Finally, we prove that the non-properness set of any real polynomial local diffeomorphism with non-dense image has codimension one.

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