Problem 6A — Is it true that any real Morsification of $f$ can be connected with one of complexity $\rho(f)$ by a g…

Definitions and mathematical statement

Let $F$ be the real versal deformation of the germ $f$ intended in the source, with base $B$, and let $\mathcal M\subset B$ be its real Morsification parameters. In the usual local interpretation, $F_u$ at such a parameter has only nondegenerate critical points in the prescribed neighborhood; a real critical point is Morse when its Hessian is nonsingular. Any further distinct-critical-value or complex-critical-point requirements must follow the missing source conventions. Let $c(u)$ be the source's complexity and $\rho(f)$ its specified optimal/target complexity, neither of which is defined in the excerpt.

The question is whether \[\forall u_0\in\mathcal M,\ \exists u_1\in\mathcal M,\ \exists\gamma:[0,1]\to B:\quad \gamma(0)=u_0,\ \gamma(1)=u_1,\ c(u_1)=\rho(f),\] with $\gamma$ generic and every Morse surgery of type $[A_2]$, traversed in the direction $0\to1$, decreasing the number of real critical points. For an ordinary real birth/death $A_2$ event, two real Morse critical points are created or annihilated. Thus, writing $r(u)$ for the number in the fixed local neighborhood, the required direction is \[r(\gamma(t_*+\eta))=r(\gamma(t_*-\eta))-2\] for sufficiently small $\eta>0$ around each such event $t_*$. Other generic events, if allowed, must obey the source's path conventions; the question is not simply existence of an arbitrary path between chambers.

The formula does not assume that $c(u)=r(u)$, nor define $\rho(f)$ as a minimum of $r$, unless that identification is confirmed from the preceding text. The germ class, base neighborhood, generic-event list, critical-point counting domain and boundary control are essential to the statement and missing here. In particular the directionality forbids inserting births at $A_2$ events even if a later cancellation lowers the final count.

Short English statement

Can every real Morsification be joined to one of the target complexity by a generic versal-deformation path using only deaths, never births, at its $A_2$ events? The source omits the definitions of complexity and $\rho(f)$.

Sources

• Local UnsolvedMath registry, permanent ID $10800015$, exact source code catalogue entry.
• Victor Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 6A: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/. Registry access 2026-07-29, html-structure extraction.
• The summary attributes complete component classifications for simple and parabolic discriminant complements to Vassiliev (2021, 2025), without full citations; it explicitly says those classifications do not settle this path question.