Currently I came across the following problem:
Say, I have an analytic function $F:\mathbb{R}^n\to\mathbb{R}$ and want to study its zero set. Everywhere, where the the gradient of $F$ does not vanish, the solution set is locally an analytic, $(n-1)$-dimensional submanifold, where the implicit function yields a local parametrization and the latter's (derivative's) explicit shape represents it as a composition of analytic functions.
I am now searching for a sufficient criterion on $F$ which implies that even around a point where the gradient of $F$ vanishes, the solution set looks like the union of (multiple) analytic manifolds.
To break it down, for me the case $n=2$ suffices (and I think possibly higher dimensions maybe studied starting from this case). I suspect, that in this case the indefiniteness of the Hessian of $F$ in a critical point $x_0$ of $F$ is such a sufficient condition. Then, I (at least think I) can show that there must be (locally around that point) four analytic solutions curves, all running into $x_0$, which are smoothly continued by one of the others. In other words, there are two solution curves, which cross in the critical point (meaning both passing that point with different derivatives), which are analytic away from $x_0$ but still smooth in that point. However, I don't know if it is analytic.
So my question again: Are there any well-known tools or theorems from differential/algebraic geometry which imply that the zero set of an analytic function is analytically continued beyond a critical point by just another branch of that zero set? Maybe weaker than my intuitive guess, or maybe even stronger, as I'm not sure whether the curves I found are analytic.
My google surveys poorly didn't result in anything helpful.
Edit: Cross-posted and answered in https://mathoverflow.net/questions/395227/when-is-a-real-analytic-variety-a-union-of-non-singular-subvarieties/396852#396852