Period Collapse in Ehrhart Quasipolynomials

Definitions and mathematical statement

Let $P\subset{\mathbb{R}}^n$ be a nonempty rational polytope, and let its denominator be \[ D(P)=\min\{D\in{\mathbb{Z}}_{>0}:DP\text{ has all vertices in }{\mathbb{Z}}^n\}. \] Its Ehrhart counting function is $L_P(t)=\#(tP\cap{\mathbb{Z}}^n)$ for $t\in{\mathbb{Z}}_{\geq0}$. It is a quasipolynomial: a positive integer $p$ is a quasiperiod if there are polynomials $q_0,\ldots,q_{p-1}\in{\mathbb{Q}}[t]$ with $L_P(t)=q_r(t)$ whenever $t\equiv r\pmod p$. Let $p(P)$ be the minimum positive quasiperiod.

The denominator is a quasiperiod and $p(P)\mid D(P)$. Thus the source's phrase "period differs from its denominator" has substantive meaning when period means the minimum one, and then it is precisely \[ p(P)<D(P), \] called period collapse. A nonminimal chosen quasiperiod would not define the intended invariant.

The problem is to give necessary or sufficient conditions on $P$ for this strict inequality, or conditions excluding it. The wording does not demand both directions of a complete characterization, and no particular dimension or class of rational polytopes is restricted. The case $D(P)=1$ cannot exhibit collapse. Lower-dimensional polytopes are included by the counting definition, without assuming constant coefficients that need extra affine-lattice hypotheses.

The supplied literature triage dated 21 August 2026 reports strong polygon results and constructions with prescribed collapse, but says its checked sources did not locate a general necessary-and-sufficient characterization of the minimum period. No exact hypotheses or construction formulas are reproduced, so these are retained as reported partial progress rather than a claimed universal period theorem.

Short English statement

Find conditions for the minimum Ehrhart quasiperiod to be a proper divisor of the rational polytope's denominator. The minimum-period convention is essential. The supplied curation reports polygon and construction results while leaving the general classification open in the checked sources.

Sources

• Local UnsolvedMath registry, permanent ID $30000110$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification reports a self-contained notation edit against the official report.
• "Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry (2005)," Oberwolfach Reports, pp. 2071--2102, DOI 10.4171/owr/2004/39: https://doi.org/10.4171/owr/2004/39. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "McAllister--Woods": https://arxiv.org/abs/math/0310255.
• Supplied "Woods": https://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r34. The other supplied source is the original report; full theorem statements are not given.
• Literature checked 21 August 2026; supplied status open, with no full characterization located in the checked sources. 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.