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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.02661 |
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| _version_ | 1866909674862804992 |
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| author | Bernstein, Daniel Irving Lundqvist, Signe |
| author_facet | Bernstein, Daniel Irving Lundqvist, Signe |
| contents | The space of \emph{parallel redrawings} of an incidence geometry $(P,H,I)$ with an assigned set of normals is the set of points and hyperplanes in $\mathbb{R}^d$ satisfying the incidences given by $(P,H,I)$, such that the hyperplanes have the assigned normals.
In 1989, Whiteley characterized the incidence geometries that have d-dimensional realizations with generic hyperplane normals such that all points and hyperplanes are distinct. However, some incidence geometries can be realized as points and hyperplanes in d-dimensional space, with the points and hyperplanes distinct, but only for specific choices of normals. Such incidence geometries are the topic of this article.
In this article, we introduce a pure condition for parallel redrawings of incidence geometries, analogous to the pure condition for bar-and-joint frameworks, introduced by White and Whiteley. The d-dimensional pure condition of an incidence geometry (P,H,I) imposes a condition on the normals assigned to the hyperplanes of (P,H,I) required for d-dimensional realizations of (P,H,I) with distinct points. We use invariant theory to show that is a bracket polynomial. We will also explicitly compute the pure condition as a bracket polynomial for some examples in the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The pure condition for incidence geometries Bernstein, Daniel Irving Lundqvist, Signe Combinatorics 52C25, 14P05 The space of \emph{parallel redrawings} of an incidence geometry $(P,H,I)$ with an assigned set of normals is the set of points and hyperplanes in $\mathbb{R}^d$ satisfying the incidences given by $(P,H,I)$, such that the hyperplanes have the assigned normals. In 1989, Whiteley characterized the incidence geometries that have d-dimensional realizations with generic hyperplane normals such that all points and hyperplanes are distinct. However, some incidence geometries can be realized as points and hyperplanes in d-dimensional space, with the points and hyperplanes distinct, but only for specific choices of normals. Such incidence geometries are the topic of this article. In this article, we introduce a pure condition for parallel redrawings of incidence geometries, analogous to the pure condition for bar-and-joint frameworks, introduced by White and Whiteley. The d-dimensional pure condition of an incidence geometry (P,H,I) imposes a condition on the normals assigned to the hyperplanes of (P,H,I) required for d-dimensional realizations of (P,H,I) with distinct points. We use invariant theory to show that is a bracket polynomial. We will also explicitly compute the pure condition as a bracket polynomial for some examples in the plane. |
| title | The pure condition for incidence geometries |
| topic | Combinatorics 52C25, 14P05 |
| url | https://arxiv.org/abs/2507.02661 |