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