Suppose I have a system of nonlinear equations. Suppose also that I'm interested in knowing if a unique solution exists. Presumably, a quick check would be to ensure that the number of equations equals the number of unknowns. If the number of equations is less than the number of unknowns (at least in linear algebra), the solution won't be unique even if it does exist, so there's no point searching for a unique solution. Is there a similar necessary condition for systems of nonlinear equations? Put another way, is it possible to create a system of nonlinear equations in $n$ unknowns that contains $m \ne n$ equations for which a unique solution exists?
EDIT: As pointed out in an answer, there are examples of trivial solutions such as: $x^2 + y^2 = 0$, in which the only solution is $x = y = 0$. I am therefore interested in nontrivial solutions.