Problem 6B — What can be said about the number $\rho(f)$?

Definitions and mathematical statement

Problem 6B asks: "What can be said about the number $\rho(f)$?" The source record does not define $\rho(f)$, the class of functions $f$, or the two bounds described in its research summary. Hence it does not identify the numerical invariant whose properties are requested.

The supplied contextual summary associates the question with two lower bounds, their possible difference and sharpness, their coincidence in corank $2$, and torsion in a relative homology group. For a real function $f:X\to{\mathbb{R}}$, the displayed pair in that summary is \[ \left(f^{-1}(( -\infty,\varepsilon]), f^{-1}(( -\infty,-\varepsilon])\right). \] With an appropriate positive $\varepsilon$, coefficient convention and domain, its relative homology is the source's "Smale group." The summary calls the associated bound (b) the Smale number. Neither the formula converting this group into that number nor the other bound is supplied. In a local singularity setting the neighborhood, regularity, isolation and endpoint-level hypotheses would also be needed; they are not inferred here.

The secondary issues named by the summary are therefore retained as unresolved contextual targets: compare the two bounds, determine whether they coincide at corank $2$, decide torsion behavior, and test sharpness of the Smale-number bound. Their exact propositions require the missing definitions. The summary reports corank-$2$ Morsification enumeration data for $\mu\le11$, without a general theorem, and explicitly assesses all these questions as open as of August 2026. This is the supplied dated literature assessment, not an independent verification.

Short English statement

Investigate the source's number $\rho(f)$, including the reported questions about two lower bounds and Smale-group torsion. The definition of $\rho$ and both bounds is missing. The supplied assessment calls the general questions unresolved as of August 2026 despite limited Morsification data.

Sources

• Local UnsolvedMath registry, permanent ID $10800016$, exact source code catalogue entry, Problem 6B, attributed to Victor Vassiliev; no proposed year supplied.
• Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 6B: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/. The background describes eighteen explicitly labeled theorem-style problem environments.
• Supplied reference [smaleR]: S. Smale, "On the structure of manifolds," Amer. J. Math. 84 (1962), 387--399, cited in the summary for the relative "Smale group." No separate references for the $\rho(f)$ computations are provided.
• HTML-structure extraction, accessed 2026-07-29. Source evidence NEEDS_REVIEW requires a current primary-literature audit. Registry status open, classification OPEN-TRIAGE; summary explicitly says open as of August 2026, with corank-$2$, $\mu\le11$ enumeration data but no general theorem. Its final arXiv-search remark is truncated and is not completed here.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.