Suppose we have a smooth function $f(x) \in \mathbb{R}^2 \rightarrow \mathbb{R}^2$, and we want to find a vector $x$ such that $f(x)=0$. Are there sufficient condition such that there exists one, or there exists a unique solution?
Looking at older threads, it looks like the answer is "no" for an arbitrary-dimensional function, but I wonder if more can be said when the nonlinear system is 2-by-2.