I have asked this before on MSE, but received no answer yet.

Say I have a set in $\mathbb{R}^n$ defined to be the zero set of an analytic function $F:\mathbb{R}^n\to\mathbb{R}^k$, $k<n$. Everywhere, where $DF$ is not singular, an analytic local parametrization is given simply by the implicit function theorem. But what about the points where $DF$ is singular?

So my question is, are there some standard tools/theorems which tell the zero set to still be an immersed manifold which just crosses/intersects itself in such a point?

More concretely, say I have an analytic function $F:\mathbb{R}^2\to\mathbb{R}$ such that $F(p)=0$ and $\nabla F(p)=0$ at some $p$. Then I can prove that, if the Hessian of $F$ in $p$ is indefinite, then there exists a neighborhood of $p$ and two smooth curves in this neigborhood intersecting only in $p$ such that $F$ vanishes on these curves. These curves have differently directed derivatives, so they indeed cross. However, while they may of course be analytically parametrized away from $p$, in $p$ itself I can only show smoothness, but no analyticity. Moreover, I cannot prove that there are only these two curves. The latter would follow from analyticity since all derivatives of multiple solution for one of the branches must coincide in all derivatives at $p$, but however, the Taylor coefficients are only implicitly given by the Taylor coefficients of $F$ and I cannot resolve it. So, since I can show for my particular $F$ that whenever $\nabla F=0$ then the Hessian is indefinite, my goal would be to state that the solution set of $F=0$ is just the union of the ranges of some analytic curves.

(1.) So, on the one hand, I'd be interested in some "standard theorem" or some "standard toolbox" to study the singular points of such $F$, which ideally tells me that I can analytically continue my analytic parametrization into the critical point.

(2.) Also I cannot figure out how this may be generalized to higher dimensions. I suspect that the higher dimensional analog of an indefinite Hessian should include an invertible Hessian to exclude $0$-eigenvalues (which in two dimensions is already granted). Probably an answer on (1.) would make an answer on (2.) redundant.

Poorly, a google research did not result in anything useful. The only text I found going in the right direction is this, approximating the solution curves by a Puiseux series expansion. However, I'm not too familiar with algebraic geometry, but I have the impression that this approach heavily relies on that there is no "second limit" if $F$ is only a polynomial.

Any help will be appreciated. Thanks!

distinctlinear factors is this: A polynomial $p(x,y)$ with real coefficients can be factored (over the reals) into linear and quadratic factors. For example, $x^3+y^3 = (x+y)(x^2-xy+y^2)$, so it has one linear factor and one quadratic factor. Meanwhile, $x^2y$ has three linear factors, but two of them are equal (i.e., not distinct). And $x^4-y^4 = (x-y)(x+y)(x^2+y^2)$ has two distinct linear factors and one quadratic factor. Indeed, your example, $x^2-y^2$ has two distinct linear factors. $\endgroup$ – Robert Bryant Jun 14 at 17:23homogeneouspolynomial $p(x,y)$ with real coefficients can be factored..." $\endgroup$ – Robert Bryant Jun 16 at 17:131more comment