Sufficient conditions for the n-dimensional real Jacobian conjecture

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2025-09-04 DOI:10.1016/j.jde.2025.113750

Changjian Liu , Yuzhou Tian

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引用次数: 0

Abstract

The real Jacobian conjecture, proposed by Randall in 1983, asserts that a polynomial map

F = (f_{1}, \dots, f_{n}) : R^{n} \to R^{n}

such that

\det D F (x) \neq 0

for all

x \in R^{n}

is injective. However, this conjecture is disproven by Pinchuk's counterexample.

This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternative 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 260 (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from

R^{2}

R^{n}

. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations 260 (2016), 5250-5258].

查看原文本刊更多论文

n维实雅可比猜想的充分条件

Randall在1983年提出的实雅可比猜想断言多项式映射F=(f1，…，fn):Rn→Rn使得det （DF(x)）对所有x∈Rn≠0是内射的。然而，这个猜想被Pinchuk的反例所推翻。本研究主要由两部分组成。首先，我们利用动力系统的定性理论给出了n维哈达玛尔定理的多项式形式的另一种证明。其次，给出了n维实雅可比猜想的几个代数充分条件。我们的结果不仅推广了[J]的主要结果。微分方程260(2016)，5250-5258]的拟齐次型，并将其从R2推广到Rn。作为我们证明过程的副产物，我们解决了Braun， gin和Llibre在[J]中提出的一个开放问题。微分方程[j].中国科学：自然科学版（2016），525 - 526。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics