What are (if there are any) the rules relating a nonlinear system of equations to the existence and number of solutions ? (i.e. the role rank-determinant of coefficient matrix play in linear systems of equations)
I've seen some numerical methods restrict themselves to "square" (same number of variables and equations) systems, but I feel this is related to the method of solution chosen rather than the existence or not and number of solutions.
I've not been able to find references on this, previous discussion here includes:
(A restricted case for two-variable polynomial equations.)
Degree theory and systems of nonlinear algebraic equations
(No Reply x3)
Conditions for existence and uniqueness of systems with linear and nonlinear equations
(Claim to be Undecidable without reference)
Existence and uniqueness of solutions to a system of non-linear equations
(follow up, no reply):
Sufficient conditions for the existence of the solution to the system of non-linear equations
Please note this is a general question, since I want a general answer, but it is also a very specific one. If anyone can point to a reference saying that you effectively can't say anything about the solutions, I'd be glad to see it.
Regards!
