On Necessary and Sufficient Conditions for the Real Jacobian Conjecture

• Yuzhou Tian ^[1] ; Yulin Zhao ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
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• Resumen

  • This paper is an expository one on necessary and sufficient conditions for the real Jacobian conjecture, which states that if F = f 1,..., f n : Rn → Rn is a polynomial map such that det DF = 0, then F is a global injective. In Euclidean space Rn, the Hadamard’s theorem asserts that the polynomial map F with det DF = 0 is a global injective if and only if F (x) approaches to infinite as x → ∞.

    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 Sabatini’s result. 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 Cima’s result on the n-dimensional real Jacobian conjecture by the n-dimensional Hadamard’s theorem