Xiuli Cen, Kangnong Hu, Yuzhou Tian
Abstract:
The real Jacobian conjecture in \(\mathbb{R}^2\) claims that if
\(F=(f,g):\mathbb{R}^2 \to \mathbb{R}^2\) is a polynomial map such that
\( \det DF(x,y)\neq 0 \) for all \((x,y)\in \mathbb{R}^2\), then \(F\) is globally
injective. However, it is known that there exists a counterexample on this conjecture.
Since then, various sufficient conditions have been proposed to ensure that the real
Jacobian conjecture holds. In this article, we generalize a result in [25]
to the quasi-homogeneous case. Moreover, we provide several examples to illustrate
the relationships between some existing sufficient conditions regarding this
conjecture.
Submitted January 15, 2026. Published February 11, 2026.
Math Subject Classifications: 34C05, 34C08, 14R15.
Key Words: Real Jacobian conjecture; Newton diagram; Bendixson compactification;
quasi-homogeneous.
DOI: 10.58997/ejde.2026.12
Show me the PDF file (2619 KB), TEX file for this article.
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Xiuli Cen School of Mathematics and Statistics HNP-LAMA, Central South University Changsha, Hunan 410083, China email: cenxiuli2010@163.com |
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Kangnong Hu School of Mathematics and Statistics HNP-LAMA, Central South University, Changsha, Hunan 410083, China email: hukangnong2000@csu.edu.cn |
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Yuzhou Tian Department of Mathematics Jinan University Guangzhou, Guangdong 510632, China email: tianyuzhou2016@163.com |
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