Integrality relations for polygonal dissections
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917777973968896 |
|---|---|
| author | Abrams, Aaron Pommersheim, Jamie |
| author_facet | Abrams, Aaron Pommersheim, Jamie |
| contents | Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. In both cases, the integrality relations are invariant under deformation of the dissection.
The trapezoid theorem implies and provides a new context for Monsky's Equidissection Theorem that a square cannot be dissected into an odd number of triangles of equal area.
A corollary of these results is that the area polynomials for parallelograms introduced in previous work have all leading coefficients equal to $\pm 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03475 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integrality relations for polygonal dissections Abrams, Aaron Pommersheim, Jamie Commutative Algebra Algebraic Geometry 52B45, 13B21 Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. In both cases, the integrality relations are invariant under deformation of the dissection. The trapezoid theorem implies and provides a new context for Monsky's Equidissection Theorem that a square cannot be dissected into an odd number of triangles of equal area. A corollary of these results is that the area polynomials for parallelograms introduced in previous work have all leading coefficients equal to $\pm 1$. |
| title | Integrality relations for polygonal dissections |
| topic | Commutative Algebra Algebraic Geometry 52B45, 13B21 |
| url | https://arxiv.org/abs/2301.03475 |