Let $f: \mathbb{R}^n \to \mathbb{R}$ be a germ of an isolated, real analytic singularity. Let $I$ be the ideal in $R = \mathbb{R}[x_1, \dots, x_n]$ generated by the components of $\nabla f$, and $Q = R/I$ the local algebra. An unfolding of the germ consists in sending $f \mapsto f+g$, where $g$ is a polynomial belonging to $Q$. The index of the germ is the Poincare-Hopf index of the singularity of the vector field $\nabla f$.

For a complex singularity, it is known that an arbitrary unfolding splits the singularity into $\mu = \dim Q$ Morse singularities. In the real case of course, this is not true: some of the singularities produced may exist only in the complex domain (e.g, $x^3 \mapsto x^3 + x$ gives a function with no real singularities).

There are a couple of constraints on which singularities may appear in the unfolding. It's obvious that any unfolding into Morse singularities can only leave an even number of them of the complex world, and the indices of those that do appear in the real world must sum to the index of $f$. Matters are complicated by the fact that an unfolding need not produce just Morse singularities, but also degenerate singularities of lower multiplicity.

**Question:** Are there any constraints on which singularities can be produced by an unfolding of $f$ other than the obvious ones I mentioned? Is there a clean way to enumerate all possible combinations of singularities produced by an unfolding, for an arbitrary function germ $f$? I would hope to be able to compute this enumeration in terms of information about $Q$ and the levels sets of $f$. I know very little about algebraic geometry, but it seems that we could perhaps relate it to deformations of the variety $f=0$.

To see what I mean, consider the germ $f = x^2y - y^3 + z^2$ of a $D^-_4$ singularity. A basis for the local algebra is $1, x, y, y^2$. Playing around for a while has convinced me there are 3 "essentially different" real unfoldings corresponding to the images below. In the first two cases a pair of Morse index 1 singularities is created, but the behaviour of the level set $f=0$ is different. In the final case we produce three Morse index 1 and one Morse index 0 singularities. It is not possible to produce just the pair with Morse index 0 and 1, and we can rule this out because the index of this combination is not right. In principle, we could produce 3 singularities, two Morse index 1 and an $A_2$ singularity that is itself comprised of the remaining Morse index 0 and 1 singularities, but if this is possible I haven't been able to figure out how to do it.

Apologies if this is a foolish question, this is not really my area.