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A Mountain Pass to the Jacobian Conjecture

Published online by Cambridge University Press:  20 November 2018

Marc Chamberland*
Dept. of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 New address: Dept. of Mathematics and Computer Science Grinnell College IA 50112 USA, email:
Gary Meisters
Dept. of Mathematics and Statistics University of Nebraska-Lincoln NE 68588-0323 USA, email:
New address: Dept. of Mathematics and Computer Science Grinnell College IA 50112 USA, email:
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This paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem 1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of ${F}'(\text{x}){F}'{{(\text{x})}^{T}}\,$ are uniformly bounded away from zero for $\text{x}\,\in \,{{\mathbb{R}}^{n}}$, where $F\,:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}^{n}}$ is a class ${{C}^{1}}$ map, then F is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class ${{C}^{1}}$ maps $F$ of ${{\mathbb{R}}^{n}}$ into itself is that all the eigenvalues of ${F}'\,(\text{x})$ are bounded away from zero on ${{\mathbb{R}}^{n}}$. This is stated as Conjecture 2.1. If true, it would imply (via Reduction-of-Degree) injectivity of polynomial maps$F\,:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}^{n}}$satisfying the hypothesis, $\det F'(x)\equiv 1$, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

Research Article
Copyright © Canadian Mathematical Society 1998


The first author was partially supported by an NSERC Post-Doctoral Fellowship.


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