A Mountain Pass to the Jacobian Conjecture.

Canadian mathematical bulletin

关键词: Injectivity;    ${cal C}^1$-maps;    polynomial maps;    Jacobian Conjecture;    Mountain Pass;   

DOI  :  10.4153/CMB-1998-058-4

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

This paper presents an approach to injectivity theorems via theMountain Pass Lemma and raises an open question. The main resultof this paper (Theorem~1.1) is proved by means of the Mountain PassLemma and states that if the eigenvalues of $F' (x)F' (x)^{T}$are uniformly bounded away from zero for $x in hbox{BbbviiR}^{n}$, where $F colon hbox{Bbbvii R}^n ightarrowhbox{Bbbvii R}^n$ is a class $cC^{1}$ map, then $F$ isinjective. This was discovered in a joint attempt by the authorsto prove a stronger result conjectured by the first author: Namely,that a sufficient condition for injectivity of class $cC^{1}$ maps$F$ of $hbox{Bbbvii R}^n$ into itself is that all the eigenvaluesof $F'(x)$ are bounded away from zero on $hbox{BbbviiR}^n$. This is stated as Conjecture~2.1. If true, it would imply(via {it Reduction-of-Degree}) {it injectivity of polynomialmaps} $F colon hbox{Bbbvii R}^n ightarrow hbox{Bbbvii R}^n${it satisfying the hypothesis}, $det F'(x) equiv 1$, of thecelebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. Thepaper ends with several examples to illustrate a variety of casesand known counterexamples to some natural questions.

【 授权许可】

Unknown