Vertical Decompositions in $\mathbb{R}^d$

Definitions and mathematical statement

Let $d\ge5$ and let $\mathcal S=\{S_1,\ldots,S_n\}$ be a family of surfaces in ${\mathbb{R}}^d$. Their arrangement subdivides the ambient space according to the surfaces. A vertical decomposition further subdivides arrangement cells by recursively introducing walls parallel to selected coordinate directions over lower-dimensional features, yielding elementary cells. The exact recursion, coordinate order, and treatment of degeneracies must be specified; the source supplies no definition of the decomposition convention.

The question asks for its complexity. Under the usual combinatorial interpretation, fix an admissible surface class $\mathcal C_d$ and a decomposition convention, let $v(\mathcal S)$ count its elementary cells or the specified total collection of faces, and seek the asymptotic order of \[ V_d(n)=\sup\{v(\mathcal S):\mathcal S\subset\mathcal C_d,\ |\mathcal S|=n\}. \] The source does not explicitly say whether it counts all cells of the full arrangement, the decomposition of a single cell or selected region, or construction time. It also does not define "surface." The research summary discusses algebraic surfaces, but gives no uniform degree or description-complexity bound. Such a restriction is essential to a complexity estimate depending only on $n$ and $d$.

The intended problem is to obtain matching general upper and lower bounds in dimensions at least five, after those inputs are fixed. Whether $d$ is treated as fixed and what dependence is allowed in constants are not explicitly stated. No exact exponent is supplied for the general case.

Short English statement

Determine the worst-case complexity of vertical decompositions of $n$ surfaces in dimension $d\ge5$. The precise surface class and whether the full arrangement or a selected cell is decomposed must be fixed.

The supplied summary says the general case is open, with tight bounds only for specialized families. Its low-dimensional and hyperplane comparisons are qualitative, including the informal phrase "$n^{d-2}$-ish," without precise hypotheses or a compatible specification of the counted object. They are not adopted as rigorous bounds for the full decomposition. The maintained TOPP entry is recorded as Open as accessed 2026-07-29; the OPEN-TRIAGE label is a supplied assessment, not independent verification.

Sources

• Local UnsolvedMath registry, permanent ID 5500019, source code catalogue entry, "Vertical Decompositions in ${\mathbb{R}}^d$." Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• The Open Problems Project, Computational Geometry, Problem 19, https://topp.openproblem.net/p19; source-TeX extraction, accessed 2026-07-29, maintained entry recorded as Open.
• Public source specification: https://github.com/edemaine/topp. The supplied rights note says its repository licence expressly covers software; problem-text reuse remains NEEDS_REVIEW. This qualification is separate from the dataset licence.
• The research summary gives no primary references or precise statements for its low-dimensional and specialized-family comparisons.