Khovanskii Bounds Beyond Simplicial Polytopes

Definitions and mathematical statement

The original statement has three linked targets: extend Khovanskii's upper-bound theorem to simplicial spheres; determine face numbers of a section $P$ from those of $Q$; and obtain corresponding bounds for general polytopes. It does not reproduce the theorem, the class of $Q$, the dimension/type of section, or the numerical inequality to extend.

For basic notation, a convex polytope $Q\subset{\mathbb{R}}^d$ is the convex hull of finitely many points. An affine section has the form $P=Q\cap A$, for a chosen affine subspace $A$, when nonempty. If $P$ has dimension $e$, write $f_i(P)$ for its number of $i$-faces, $0\leq i<e$, and $f(P)$ for this face vector. A simplicial sphere is a finite simplicial complex whose realization is a sphere in a specified topological/PL/homological convention; the source does not choose among those variants.

The source program can therefore be recorded as seeking:

1. a simplicial-sphere version of the specified Khovanskii inequality, once its actual statement and sphere analogue are fixed;
2. sharp bounds or a characterization of possible $f(P)$ in terms of $f(Q)$ and the necessary section data;
3. an extension beyond the intended initial simplicial-polytope class to general convex polytopes.

"Determine from" may mean exact determination, extremal bounds, or an attainable range; these differ when section position varies. Section dimension, genericity, centrality, and incidence assumptions cannot be omitted from an explicit theorem. An abstract simplicial sphere has no canonical affine section without an additional realization or combinatorial construction. No missing inequality or construction is invented here.

The supplied literature assessment dated 21 August 2026 describes a heading about clique complexes and the Charney--Davis conjecture, contradicting the provided Khovanskii statement. Its cited subdivision/status sources therefore cannot be assumed to establish the status of these three targets. The registry labels the record open, but the mismatch and missing theorem hypotheses remain unresolved source issues rather than grounds for silently replacing the question by Charney--Davis.

Short English statement

The statement asks for Khovanskii-type bounds for sphere analogues, polytope sections, and general polytopes. Its exact starting theorem and section data are missing. The supplied Charney--Davis assessment concerns a different topic and is retained as an explicit curation mismatch.

Sources

• Local UnsolvedMath registry, permanent ID $30000088$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification reports a self-contained edit against the official report, though the starting inequality is not reproduced.
• "Combinatorial Commutative Algebra (2005)," Oberwolfach Reports, pp. 1703--1746, DOI 10.4171/owr/2004/32: https://doi.org/10.4171/owr/2004/32. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "Frohmader, Charney--Davis for subdivisions": https://arxiv.org/abs/0806.4213.
• Supplied "Current status discussion": https://mathoverflow.net/questions/349406/updates-to-stanleys-1999-survey-of-positivity-problems-in-algebraic-combinatori. These Charney--Davis references are supplied but their relevance to the stated Khovanskii targets is unverified; no complete citation of Khovanskii's theorem is given.
• Literature checked 21 August 2026; supplied status open, with an assessment about a different apparent excerpt. Registry created 25 August 2026 and updated 16 September 2026. This dated triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.