Combinatorial Nonnegativity for Polytope $S$-Polynomials

Definitions and mathematical statement

Let $M$ be a lattice, meaning a free abelian group of finite rank, and let $M_{{\mathbb{R}}}=M\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. A nonempty lattice polytope is the convex hull of a finite nonempty subset of $M$ in $M_{{\mathbb{R}}}$. The supplied statement gives no restriction on dimension and imposes no smoothness, simplicity, or full-dimensionality hypothesis.

For a lattice polytope $P$, write $S_P(t)$ for the polynomial called its $S$-polynomial in the supplied source. This is a placeholder for the source's invariant: the supplied record contains no definition or normalization of $S_P$. If its coefficient expansion is written as \[ S_P(t)=\sum_{j\geq 0}s_j(P)t^j, \] with only finitely many nonzero coefficients, the requested statement is the following methodological problem:

For every lattice polytope $P$, find a combinatorial proof that $s_j(P)\geq 0$ for every $j\geq 0$.

The request concerns a combinatorial proof of coefficient nonnegativity. The supplied dated assessment describes nonnegativity and the cited polynomial identities as already established; it labels the request for a genuinely combinatorial proof as open. These are supplied assessments, not independently verified conclusions of this draft.

The formulation remains incomplete until the intended $S$-polynomial and its normalization are specified. The record also gives no criterion for what qualifies as a combinatorial proof and does not specify an empty-polytope convention. Consequently, the displayed expansion expresses the requested coefficient condition but does not define the missing invariant.

Short English statement

The problem asks for a combinatorial proof that every coefficient of the $S$-polynomial of every lattice polytope is nonnegative. The supplied curation treats nonnegativity as established and the requested proof method as open. A precise standalone formulation still requires the source's definition of the $S$-polynomial and clarification of the intended meaning of a combinatorial proof.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002043$, title: Combinatorial Nonnegativity for Polytope $S$-Polynomials; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and supplies proposed year $2012$, with no named proposer.
• Supplied report citation: Toric Geometry (2013), Oberwolfach Reports, DOI $10.4171/owr/2012/21$, pp. 1231--1306. https://doi.org/10.4171/owr/2012/21. The record attributes the problem to workshop open-problem discussions documented in this report. Its verification field states that the official report was checked and the complete statement preserved with normalized notation; this is supplied verification metadata, not a source visit performed for this draft.
• Supplied literature reference: Batyrev--Nill, Combinatorial aspects of mirror symmetry. https://www.researchgate.net/publication/2207724_Combinatorial_aspects_of_mirror_symmetry.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the cited $S$-polynomial identities and nonnegativity as established, and reports that no authoritative resolution of the combinatorial-proof request was located. This is supplied dated curation derived from the Oberwolfach report, not independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.