Consider an arbitrary system of non-linear equations $F(x)=0$ where $F:\mathbb{R}^n \rightarrow \mathbb{R}^m$. Are there any properties to check in order to study whether solutions exist, are unique and, more generally, what theories could one use in order to characterize sets of solutions in the case of non-uniqueness? Any references or hints are appreciated.
As a note, I am aware of the implicit function theorem and the use of Jacobians to study solutions locally.
In general this is undecidable. The tools you mention (implicit function theorem and Jacobians) are quite useful locally, especially for the case $m=n$ when the Jacobian is nonzero at most points, but unless you have good control on the behaviour of $F(x)$ as $x$ gets large it may be impossible to tell whether there is a solution somewhere "out there". For example, suppose you have a system of polynomial equations in variables $x_j$. Include the equations $\sin(\pi x_j) = 0$ and you force the variables in a solution to be integers. But there is no algorithm to decide whether a system of polynomials has integer solutions (Hilbert's 10th problem).
In the case of a polynomial system, tools of algebraic geometry (Grobner bases, characteristic sets, etc) can be used.
