On polynomials associated to Voronoi diagrams of point sets and crossing numbers
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910685883006976 |
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| author | Claverol, Mercè Heras-Parrilla, Andrea de las Flores-Peñaloza, David Huemer, Clemens Orden, David |
| author_facet | Claverol, Mercè Heras-Parrilla, Andrea de las Flores-Peñaloza, David Huemer, Clemens Orden, David |
| contents | Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On polynomials associated to Voronoi diagrams of point sets and crossing numbers Claverol, Mercè Heras-Parrilla, Andrea de las Flores-Peñaloza, David Huemer, Clemens Orden, David Combinatorics Computational Geometry Discrete Mathematics Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials. |
| title | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| topic | Combinatorics Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2304.12238 |