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arXiv · 2610.33875

Global invertibility of polynomial mappings and connectedness of preimages of large spheres

Francisco Braun·Zbigniew Jelonek·Maria Michalska
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Abstract

We prove that a polynomial mapping $\mathbb{R}^2 \to \mathbb{R}^2$ with nowhere zero Jacobian determinant is a global diffeomorphism provided the preimage of large circles are connected. In higher dimensions this result fails and we present families of counterexamples. Yet, in dimension $3$, we prove that if the preimages of large spheres are connected, then the Jelonek set of the mapping is not homeomorphic to a line and, provided that it has codimension at least $2$, cannot have a single end at infinity.

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Francisco Braun, Zbigniew Jelonek, Maria Michalska. 2026-10-24. Global invertibility of polynomial mappings and connectedness of preimages of large spheres. https://arxiv.org/abs/2610.33875

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