Spaces of polygonal triangulations and Monsky polynomials
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912457229860864 |
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| author | Abrams, Aaron Pommersheim, James |
| author_facet | Abrams, Aaron Pommersheim, James |
| contents | Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation $\T$ of a square must satisfy a single irreducible homogeneous polynomial relation $p(\T)$ depending only on the combinatorics of $\T$. The invariant $p(\T)$ is called the \emph{Monsky polynomial}; it captures algebraic, geometric, and combinatorial information about $\T$. We give an algorithm that computes a lower bound on the degree of $p(\T)$, and we present several examples in which the algorithm is used to compute the degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23444 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spaces of polygonal triangulations and Monsky polynomials Abrams, Aaron Pommersheim, James Algebraic Geometry Combinatorics Metric Geometry Given a combinatorial triangulation of an $n$-gon, we study (a) the space of all possible drawings in the plane such the edges are straight line segments and the boundary has a fixed shape, and (b) the algebraic variety of possibilities for the areas of the triangles in such drawings. We define a generalized notion of triangulation, and we show that the areas of the triangles in a generalized triangulation $\T$ of a square must satisfy a single irreducible homogeneous polynomial relation $p(\T)$ depending only on the combinatorics of $\T$. The invariant $p(\T)$ is called the \emph{Monsky polynomial}; it captures algebraic, geometric, and combinatorial information about $\T$. We give an algorithm that computes a lower bound on the degree of $p(\T)$, and we present several examples in which the algorithm is used to compute the degree. |
| title | Spaces of polygonal triangulations and Monsky polynomials |
| topic | Algebraic Geometry Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2506.23444 |