Boundaries of pseudointegral polygons
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915100424667136 |
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| author | McAllister, Tyrrell B. Williford, Jason S. |
| author_facet | McAllister, Tyrrell B. Williford, Jason S. |
| contents | We prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most $9$ lattice points on its boundary, where a polygon $P$ is called pseudointegral if the Ehrhart function of $P$ is a polynomial. We further show that such a triangle never has exactly $7$ lattice points on its boundary. Our results determine the set of all Ehrhart polynomials of rational triangles with one interior lattice point.
In addition, we construct convex pseudointegral polygons with $i$ interior lattice points and $b$ boundary lattice points for all positive integral values of $(i,b)$ such that $b \le 5i + 4$. This is in contrast to integral polygons, which must satisfy $b \le 2i + 7$ by a result of Scott. Our constructions yield many new Ehrhart polynomials of rational polygons in the $i \ge 2$ case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundaries of pseudointegral polygons McAllister, Tyrrell B. Williford, Jason S. Combinatorics 52C05, 11H06, 05A15, 52B45 We prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most $9$ lattice points on its boundary, where a polygon $P$ is called pseudointegral if the Ehrhart function of $P$ is a polynomial. We further show that such a triangle never has exactly $7$ lattice points on its boundary. Our results determine the set of all Ehrhart polynomials of rational triangles with one interior lattice point. In addition, we construct convex pseudointegral polygons with $i$ interior lattice points and $b$ boundary lattice points for all positive integral values of $(i,b)$ such that $b \le 5i + 4$. This is in contrast to integral polygons, which must satisfy $b \le 2i + 7$ by a result of Scott. Our constructions yield many new Ehrhart polynomials of rational polygons in the $i \ge 2$ case. |
| title | Boundaries of pseudointegral polygons |
| topic | Combinatorics 52C05, 11H06, 05A15, 52B45 |
| url | https://arxiv.org/abs/2501.00667 |