In linear systems of equations, it is well-known that the independence of equations $$Ax = b$$ can be determined by checking whether the determinant of the matrix $A$ is nonzero.
I am interested in finding similar "simple" criteria for testing the independence of conditions in nonlinear systems expressed as:
$$\begin{cases} \phi^1(x,y,z) = 0 \\ \phi^2(x,y,z) = 0 \\ \phi^3(x,y,z) = 0 \end{cases}$$
Are there any straightforward methods or conditions that can be used to ascertain whether a set of nonlinear equations constitutes an independent system? I would prefer criteria that can be easily verified or computed without relying on complex numerical techniques.
I am aware that one possible approach involves the Jacobian (the determinant of the Jacobian matrix) $\det\frac{\partial \phi^i}{\partial x^j}$. However, this criterion can sometimes yield incorrect results. For instance, consider the following system:
$$\begin{cases}x^2 - y^2 = 0 \\ x^2 - z^2 = 0 \\ (y^2 - z^2)\sin z = 0 \end{cases}$$
It is evident that this system possesses an entire family of solutions with $|x|=|y|=|z|$ (for $x,y,z \in \mathbb{R}$). However, the Jacobian evaluates to $4 \left(y^3 z \cos (x)-y z^3 \cos (x)\right)$, which is nonzero.
I would greatly appreciate any insights and references. Thank you!