You might be interested by C. Soulé, M.Kaufman, R.Thomas [results][1] (search "multistationarity"). This might seem unrelated to your question but it is in fact related.

Briefly, they study various *necessary* conditions for a differential equation $dx/dt=F(x)$, $x\in\mathbb{R}^n$ to have *several* non degenerate stationary points $F(x)=0$. 

The conditions depend on a signed "interaction graph" $G(x)$ deduced from the signs in the Jacobian matrix of $F$ at $x$. 

By taking the contrapositive, applied to $F-c$ or various other simple transform , you obtain *sufficient* conditions for $F$ to be injective (assuming non-vanishing jacobian determinant).

Hope this helps.

  [1]: http://www.ihes.fr/~soule/documents/publications.html