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Sufficient conditions for the n-dimensional real Jacobian conjecture

Changjian Liu, Yuzhou Tian

Abstract

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].

Sufficient conditions for the n-dimensional real Jacobian conjecture

Abstract

The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if is a polynomial map such that for all , then 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 -dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the -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 to . 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].

Paper Structure

This paper contains 4 sections, 16 theorems, 80 equations.

Table of Contents

  1. Introduction and main results
  2. A dynamical proof of Theorem \ref{['th3']}
  3. Algebraic conditions
  4. Proof of Theorems \ref{['th13']} and \ref{['th11']}

Key Result

Theorem A

(see MR3448779) Assume that the polynomial map $F=\left(f,g\right):\mathbb{R}^2\rightarrow \mathbb{R}^2$ satisfies $F\left(0,0\right)=\left(0,0\right)$ and $\det DF\left(x,y\right)\neq0$ for all $\left(x,y\right)\in\mathbb{R}^2$. If the higher homogeneous terms of the polynomials $ff_x+gg_x$ and $ff

Theorems & Definitions (28)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 1
  • Theorem 5
  • Lemma 1
  • ...and 18 more