Suppose that we want to find out if the following system of equations have a unique solution: $$ g_i(x)=0 \quad i=1,2,\ldots,m, $$ where $x \in {\Bbb R}^n$ and $g_i(x)$ is any arbitrary function $g_i: {\Bbb R}^n \rightarrow {\Bbb R} $.
In general, showing that a nonlinear system of equations has a unique solution can be hard.
Consider the following optimization problem:
$$ \min\ \sum_i g_i(x)^2 $$
If the objective function turns out to be convex, then can we conclude that the original system of nonlinear equations has at most one unique solution?
Short answer is NO. However if the objective is strictly convex the system has atmost one solution.