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 (Discrete mathematics and theoretical computer science, 2024)
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 $E_{≤ k}$ 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.