I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.

In 1953, [Samuelson][1] asked the following:

>If the upper left-hand principal minors of the Jacobian matrix of a map $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ do not vanish, is it true that $F$ must be injective?

In 1965, [Gale][2] and Nikaido gave a counterexample in $\mathbb{R}^2$. In [their paper][3] the following is proved

>**Gale-Nikaido theorem:** If all the principal minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ are **positive**, then $F$ is injective. 

Since then, some effort has been made to weaken the assumption in Gale-Nikaido theorem since the assumption seems to be too restrictive in application. A comprehensive dicussion can be found in T. Parthasarathy, *On Global Univalence Theorems*, Lecture Notes in Mathematics, Vol. 977, 1983. In the case of polynomial map, this is related to the real version of [Jacobian conjecture][4].

A possible generalization I'm interested in is the following, which seems to be open.

>**Question:** If all the principal minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ **do not vanish**, is $F$ necessarily injective?

In Gale and Nikaido's paper, the case of $\mathbb{R}^2$ was answered in affirmative, the case of $\mathbb{R}^3$ was claimed in affirmative (yet no complete proof seems to be known).

My motivation comes from trying to make a change of variables to globally rectify a curved coordinate system so that Plancherel theorem can be applied. Any information would be appreciated : )


  [1]: http://en.wikipedia.org/wiki/Paul_Samuelson
  [2]: http://en.wikipedia.org/wiki/David_Gale
  [3]: http://www.springerlink.com/content/w007438g2kw60qjt/
  [4]: http://en.wikipedia.org/wiki/Jacobian_conjecture