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arXiv · 2304.12238

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Mercè Claverol·Andrea de las Heras-Parrilla·David Flores-Peñaloza·Clemens Huemer·David Orden
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Abstract

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.

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Mercè Claverol, Andrea de las Heras-Parrilla, David Flores-Peñaloza, Clemens Huemer, David Orden. 2023-04-24. On polynomials associated to Voronoi diagrams of point sets and crossing numbers. https://doi.org/10.46298/dmtcs.12443

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