Unique solution of nolinear equation set

Question

$$\left\{ \begin{aligned} f_1(x_1,x_2...x_n)=0 \\ f_2(x_1,x_2...x_n)=0 \\ \vdots \\ f_n(x_1,x_2...x_n)=0 \end{aligned} \right. $$ $f_i\in C^\infty(R^n)$,what is the condition that make the equation set has unique solution? In fact ,I think the $f_i\in C^\infty(R^n)$ can be changed to $f_i\in C^2(R^n)$.

Answer 1

One sufficient condition is that $f$ is injective. By the inverse function theorem, it suffices to prove that the differential $Df = (\displaystyle \frac{\partial f_i}{\partial x_j})$ is invertible. In fact, you only need that $f_i$ are $C^1$, and the differential is invertible.

As noted in the comments, this shows that the system has at most one solution. It does not imply that a solution exists.