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arXiv · 2605.12302

A sharp degree bound in the real Jacobian conjecture

F. Braun·J. Gwoździewicz·F. Fernandes·B. Oréfice-Okamoto
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Abstract

Let $F=(p,q):\mathbb R^2\to \mathbb R^2$ be a polynomial map with nowhere zero Jacobian determinant. A long-standing problem is to determine the largest integer $k$ such that the condition $°p\le k$ guarantees the global injectivity of $F$. Although several partial results have been obtained over the past $30$ years, the sharp degree bound has remained unknown. In this paper, we prove that $F$ is injective whenever $°p=6$. On the other hand, we construct a non-injective polynomial map with nowhere vanishing Jacobian determinant for which $°p=7$. Combined with the previously known injectivity results for $°p\le 5$, our results completely settle the problem and establish the optimal degree bound. More precisely, we show that $7$ is the minimal degree for which non-injective examples can occur.

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F. Braun, J. Gwoździewicz, F. Fernandes, B. Oréfice-Okamoto. 2026-05-24. A sharp degree bound in the real Jacobian conjecture. https://arxiv.org/abs/2605.12302

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