Counting half-integral polygons with collinear interior lattice points
A half-integral polygon is a two-dimensional convex polygon $P \subset \mathbb{R}^2$ with vertices in $\tfrac12 \mathbb{Z}^2$; integral polygons are included. Two polygons are equivalent when $P' = UP + b$ for some $U \in \operatorname{GL}_2(\mathbb{Z})$ and $b \in \mathbb{Z}^2$. Bohnert and Springer show that a polygon with vertices in $\tfrac1k \mathbb{Z}^2$ whose interior lattice points are collinear and more than $k$ in number is equivalent to a polygon in $\mathbb{R} \times [-1, 1]$, and they classified the half-integral polygons with $i$ collinear interior lattice points by computer for $3 \le i \le 20$.
Problem. For an integer $i \ge 3$ let $N(i)$ be the number of equivalence classes of half-integral polygons with exactly $i$ interior lattice points, all on one line. Is $$N(i) = \frac{i+1}{1260}\left(512i^6 + 12928i^5 + 137740i^4 + 685145i^3 + 1582743i^2 + 1665222i + 710640\right)$$ for every $i \ge 3$?
Source: Bohnert and Springer (2026), Conjecture 3.7.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, lattice polygons, rational polygons
References
@article{bohnert2026,
author = {Bohnert, Martin and Springer, Justus},
title = {Classifying rational polygons with small denominator and few interior lattice points},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {2},
year = {2026},
pages = {1238--1265},
doi = {10.1007/s00454-026-00822-0},
eprint = {2410.17244},
archivePrefix = {arXiv},
}
@misc{springer2024data,
author = {Springer, Justus and Bohnert, Martin},
title = {Half-integral polygons with few interior lattice points},
publisher = {Zenodo},
year = {2024},
version = {1.0.0},
doi = {10.5281/zenodo.13928298},
note = {Dataset},
}
@misc{aim-half-integral-collinear-count,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Counting half-integral polygons with collinear interior lattice points}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/571-half-integral-collinear-count.md},
note = {Entry 571 at commit 24edaad, 2026-10-15},
}
Resolution claims
We count convex half-integral polygons with a prescribed number of collinear interior lattice points, up to affine unimodular equivalence. For every $i\geq3$, the count is the degree-seven polynomial conjectured by Bohnert and Springer. The proof is computer-assisted: their strip theorem reduces the polygons to two boundary chains on five integer rows, and a unique integer shear normalization gives finitely many …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00797, "Counting half-integral polygons with collinear interior lattice points", clopen.solutions.