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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.20846 |
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| _version_ | 1866909367929929728 |
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| author | Liu, Changjian Tian, Yuzhou |
| author_facet | Liu, Changjian Tian, Yuzhou |
| contents | The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if $F=\left(f_1,\ldots ,f_n\right):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial map such that $\det DF\left(\mathbf{x}\right)\neq0$ for all $\mathbf{x}\in\mathbb{R}^n$, then $F$ is injective.
This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternate proof of the polynomial version of the $n$-dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the $n$-dimensional real Jacobian conjecture. Our results not only extend the main result of [J. Differential Equations {\bf 260} (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from $\mathbb{R}^2$ to $\mathbb{R}^n$. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations {\bf 260} (2016), 5250-5258]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for the n-dimensional real Jacobian conjecture Liu, Changjian Tian, Yuzhou Dynamical Systems The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if $F=\left(f_1,\ldots ,f_n\right):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial map such that $\det DF\left(\mathbf{x}\right)\neq0$ for all $\mathbf{x}\in\mathbb{R}^n$, then $F$ is injective. This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternate proof of the polynomial version of the $n$-dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the $n$-dimensional real Jacobian conjecture. Our results not only extend the main result of [J. Differential Equations {\bf 260} (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from $\mathbb{R}^2$ to $\mathbb{R}^n$. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations {\bf 260} (2016), 5250-5258]. |
| title | Sufficient conditions for the n-dimensional real Jacobian conjecture |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.20846 |