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PreprintCC BY 4.0arXiv

Enumerating Integral Points on Triangles in ℝ3

  • Max Aldrich
  • Daniel Gulbrandsen
  • Clara Keddy
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Abstract

We present two ways of constructing Ehrhart polynomials corresponding to certain triangles in ℝ3. One of our methods employs Popoviciu's Theorem, while the other method uses a version of Pick's Theorem adapted to rank-2 lattices in ℝ3. We then apply our methods to linear Diophantine equations of the form ax+by+cz=n and develop upper and lower bounds on the number of solutions in nonnegative integers x, y, and z. The bounds we develop are closed form quasi-polynomials, and the difference between them is bounded by a constant depending only on a and b. As an application, we prove that the kth Frobenius number exists for any triple of positive integers and any k.

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Cite

BibTeXRIS
BibTeX
@misc{aldrich2026enumerating,
  title = {{Enumerating Integral Points on Triangles in $\mathbbR^3$}},
  author = {Max Aldrich and Daniel Gulbrandsen and Clara Keddy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20899},
  eprint = {2610.20899},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20899},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Enumerating Integral Points on Triangles in ℝ^3
AU  - Max Aldrich
AU  - Daniel Gulbrandsen
AU  - Clara Keddy
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.20899
UR  - https://arxiv.org/abs/2610.20899
AB  - We present two ways of constructing Ehrhart polynomials corresponding to certain triangles in ℝ^3. One of our methods employs Popoviciu's Theorem, while the other method uses a version of Pick's Theorem adapted to rank-2 lattices in ℝ^3. We then apply our methods to linear Diophantine equations of the form ax+by+cz=n and develop upper and lower bounds on the number of solutions in nonnegative integers x, y, and z. The bounds we develop are closed form quasi-polynomials, and the difference between them is bounded by a constant depending only on a and b. As an application, we prove that the kth Frobenius number exists for any triple of positive integers and any k.
ER  -