Necessary condition for uniqueness solution in a system of non-linear equations

Question

Consider a system of non-linear equations with $n$ equations and $m$ unknowns. Is $m=n$ a necessary condition for having one unique solution?

Answer 1

Assuming you're talking about solutions in the real numbers, you can have a unique solution with just one equation and any number of unknowns, e.g. for $$ x_1^2 + x_2^2 + \ldots + x_n^2 = 0$$

Answer 2

Not necessarily. Consider a system with $m = n = 2$, with two equations $f(x, y) = 0$ and $g(x, y) = 0$, and one unique solution. If we add another equation $h(x, y) = cf(x, y)$ for some non-zero constant $c$, we have a system with $m = 2$ and $n = 3$. However, the solution set is exactly the same, and thus this system still only has one unique solution.

Answer 3

No. Even with $m=1, n=2.$ For example $(x^2-2 x+1=0)\land (x^2-3 x+2=0).$