You might be interested by C. Soulé, M.Kaufman, R.Thomas results (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.

You might be interested by C. Soulé, M.Kaufman, R.Thomas results (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.