A sharp degree bound in the real Jacobian conjecture
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866914596204314624 |
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| author | Braun, F. Gwoździewicz, J. Fernandes, F. Oréfice-Okamoto, B. |
| author_facet | Braun, F. Gwoździewicz, J. Fernandes, F. Oréfice-Okamoto, B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A sharp degree bound in the real Jacobian conjecture Braun, F. Gwoździewicz, J. Fernandes, F. Oréfice-Okamoto, B. Algebraic Geometry Commutative Algebra 14R15, 14H20, 37C10, 34C05 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. |
| title | A sharp degree bound in the real Jacobian conjecture |
| topic | Algebraic Geometry Commutative Algebra 14R15, 14H20, 37C10, 34C05 |
| url | https://arxiv.org/abs/2605.12302 |