Counting Ehrhart quasipolynomials of half-integral polygons
For a two-dimensional convex polygon $P$ with vertices in $\tfrac12 \mathbb{Z}^2$ let $$L_P(t) = |tP \cap \mathbb{Z}^2| \qquad (t \in \mathbb{Z}_{\ge 0})$$ be its Ehrhart quasipolynomial; its coefficients have period dividing two. Integral polygons, and polygons whose quasipolynomial is a polynomial (period collapse), are included. For a lattice polygon with $i$ interior and $b$ boundary lattice points, $L_P(t) = At^2 + \frac b2 t + 1$ with $A$ determined by $i$ and $b$ through Pick's formula, and Scott's inequality $b \le 2i + 6$ for $i \ge 2$, together with $b \ge 3$ and the fact that every intermediate value of $b$ occurs, gives exactly $2i + 4$ distinct Ehrhart polynomials of lattice polygons with $i \ge 2$ interior lattice points. Bohnert and Springer computed the number of distinct Ehrhart quasipolynomials of half-integral polygons with $i$ interior lattice points for $i \le 16$.
Problem. For an integer $i \ge 2$ let $E(i)$ be the number of distinct functions $L_P$ over such polygons $P$ with exactly $i$ interior lattice points and at least two boundary lattice points; polygons with the same function count once, and no condition is placed on the position of the interior lattice points. Is $$E(i) = \frac92 i^3 + 36 i^2 + \frac{175}{2} i + 53$$ for every $i \ge 2$?
Source: Bohnert and Springer (2026), Conjecture 6.8.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, Ehrhart quasipolynomials, 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{bohnert2024scott,
author = {Bohnert, Martin and Springer, Justus},
title = {Generalizations of {S}cott's inequality and {P}ick's formula to rational polygons},
year = {2024},
eprint = {2411.11187},
archivePrefix = {arXiv},
primaryClass = {math.CO},
}
@article{scott1976,
author = {Scott, P. R.},
title = {On convex lattice polygons},
journal = {Bull. Austral. Math. Soc.},
volume = {15},
number = {3},
year = {1976},
pages = {395--399},
doi = {10.1017/S0004972700022826},
}
@misc{aim-half-integral-ehrhart-count,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Counting Ehrhart quasipolynomials of half-integral polygons}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/572-half-integral-ehrhart-count.md},
note = {Entry 572 at commit 24edaad, 2026-10-15},
}
Resolution claims
Fix an integer \(i\ge2\). We count the distinct Ehrhart functions of convex half-integral polygons with exactly \(i\) interior lattice points and at least two boundary lattice points. The number is \((9i^3+72i^2+175i+106)/2\), as proposed by Bohnert and Springer. We describe the complete set of functions by their area and their boundary counts before and after doubling. Convex section estimates isolate two …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00798, "Counting Ehrhart quasipolynomials of half-integral polygons", clopen.solutions.