On necessary and sufficient conditions for the real Jacobian conjecture. (English) Zbl 1529.58004
Summary: This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if \(F = \left(f^1, \dots, f^n\right): \mathbb{R}^n\rightarrow\mathbb{R}^n\) is a polynomial map such that \(\det DF\neq 0\), then \(F\) is a global injective. In Euclidean space \(\mathbb{R}^n\), the Hadamard’s theorem asserts that the polynomial map \(F\) with \(\det DF\neq 0\) is a global injective if and only if \(\|F(\mathbf{x})\|\) approaches to infinite as \(\|\mathbf{x}\|\rightarrow\infty\). The first part of this paper is to study the two-dimensional real Jacobian conjecture via Bendixson compactification, by which we provide a new version of M. Sabatini’s result [Nonlinear Anal., Theory Methods Appl. 34, No. 6, 829–838 (1998; Zbl 0949.34018)]. This version characterizes the global injectivity of polynomial map \(F\) by the local analysis of a singular point (at infinity) of a suitable vector field coming from the polynomial map \(F\). Moreover, applying the above results we present a dynamical proof of the two-dimensional Hadamard’s theorem. In the second part, we give an alternative proof of the A. Cima et al.’s result [Nonlinear Anal., Theory Methods Appl. 26, No. 4, 877–885 (1996; Zbl 0845.58014)] on the \(n\)-dimensional real Jacobian conjecture by the \(n\)-dimensional Hadamard’s theorem.
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