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

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 $S$ enclo...

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Bibliographic Details
Published in:	Discrete mathematics and theoretical computer science Vol. 26:2; no. Combinatorics
Main Authors:	Claverol, Mercè, Heras-Parrilla, Andrea de las, Flores-Peñaloza, David, Huemer, Clemens, Orden, David
Format:	Journal Article
Language:	English
Published:	Discrete Mathematics & Theoretical Computer Science 01.01.2024
Subjects:	computer science - computational geometry computer science - discrete mathematics mathematics - combinatorics
ISSN:	1365-8050, 1365-8050
Online Access:	Get full text
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Summary:	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.
ISSN:	1365-8050 1365-8050
DOI:	10.46298/dmtcs.12443