Title: On polynomials associated to Voronoi diagrams of point sets and crossing numbers
Domain: Math

Question: How can one develop a rigorous framework that connects Voronoi diagrams, circle polynomials, and $(\leq k)$-edge polynomials of point sets to the rectilinear crossing number? In particular, what structural relations among these polynomials allow the derivation of exact formulas for the crossing number through their coefficients and roots, and how can these results be used to characterize when all roots lie on the unit circle and to establish bounds on their location for general point sets?

Background: | | * Recall that the **rectilinear crossing number** $cr(S)$ of a point set $S$ is the number of convex quadrilaterals or equivalently the number of edge crossings in the straight-line drawing of $K_n$ on $S$. | * Known results establish relations between **circle counts**, **Voronoi diagram vertices**, and **$(\leq k)$-edges**, which connect combinatorial geometry with crossing numbers. | * Classical approaches: inequalities for $c_k$, relations $v_k = c_{k-1} + c_{k-2}$, and formulas linking sums of $k\cdot c_k$ to binomial coefficients plus crossing numbers. | * Understanding root distributions of polynomials derived from these combinatorial quantities (Voronoi, circle, and edge polynomials) is important for analyzing extremal properties of $cr(S)$. |

Definitions: * **Voronoi Polynomial**: $p_V(z) = \sum_{k=1}^{n-1} v_k z^{k-1}$, where $v_k$ counts vertices in the order-$k$ Voronoi diagram. | * **Circle Polynomial**: $p_C(z) = \sum_{k=0}^{n-3} c_k z^k$, where $c_k$ counts circles through three points enclosing exactly $k$ others. | * **$(\leq k)$-edge Polynomial**: $p_E(z) = \sum_{k=0}^{n-3} E_{\leq k} z^k$, with $E_{\leq k}$ the number of edges with at most $k$ points to one side. | * **Reciprocal Polynomial**: For a polynomial $P(z) = \sum c_k z^k$, the reciprocal is $P^*(z) = \sum c_k z^{n-k}$, often used in root symmetry arguments. | * **Convex Position Characterization**: All roots of $p_V(z)$ lie on the unit circle if and only if the point set is in convex position.

Methodology: * Apply **Campbell’s theorem** and combinatorial identities to link coefficients of the polynomials to crossing numbers. | * Use **derivatives at $z=1$** to extract expressions involving $cr(S)$. | * Employ **Aziz–Mohammad identity** to connect polynomial coefficients and roots with crossing number formulas. | * Use **Palindromic polynomial root theorems** to characterize unit circle root distributions in convex configurations. | * Apply **Laguerre’s separation theorem** and **Obrechkoff’s theorem** to bound root locations for non-convex configurations. | * Leverage **Eneström–Kakeya theorem** and **analytic bounds** (e.g., Titchmarsh, Rahman–Schmeisser) to estimate moduli of roots of $p_E(z)$.

Answer: For a set $S$ of $n$ points in general position in the plane, the rectilinear crossing number $cr(S)$, which counts the number of pairwise edge crossings in the complete graph $K_n$ when drawn with straight lines, is closely related to three key polynomials associated with Voronoi diagrams and geometric properties of the point set: | | 1. **Voronoi Polynomial** | The Voronoi polynomial $p_V(z)$ is defined as: | | $$ | p_V(z) = \sum_{k=1}^{n-1} v_k z^{k-1}, | $$ | | where $v_k$ represents the number of vertices in the order-$k$ Voronoi diagram of the point set $S$. The vertices of the Voronoi diagram are linked to the combinatorics of point sets, where the coefficients of this polynomial provide insights into the geometric arrangement of the points in $S$. | | 2. **Circle Polynomial** | The circle polynomial $p_C(z)$ is defined as: | | $$ | p_C(z) = \sum_{k=0}^{n-3} c_k z^k, | $$ | | where $c_k$ counts the number of circles passing through three points of $S$ and enclosing exactly $k$ other points of $S$. The relationship between the circle polynomial and the Voronoi diagram is critical in understanding the spatial structure of the points and their influence on the rectilinear crossing number. | | 3. **$(\leq k)$-Edge Polynomial** | The $(\leq k)$-edge polynomial $p_E(z)$ is defined as: | | $$ | p_E(z) = \sum_{k=0}^{n-3} E_{\leq k} z^k, | $$ | | where $E_{\leq k}$ counts the number of edges in the set $S$ that have at most $k$ points on one side. This polynomial encodes higher-order geometric properties of the set, linking the arrangement of edges to the combinatorial structure of the Voronoi diagram. | | ### **Mathematical Insights and Contributions**: | | * **Crossing Number Expression**: The rectilinear crossing number $cr(S)$ can be derived from the derivatives of the polynomials $p_V(z)$, $p_C(z)$, and $p_E(z)$ evaluated at $z = 1$: | | * For $p_C(z)$: | | $$ | p'_C(1) = \binom{n}{4} + cr(S) | $$ | * For $p_V(z)$: | | $$ | p'_V(1) = \binom{n}{3} + 2 \binom{n}{4} + 2cr(S) | $$ | * For $p_E(z)$: | | $$ | p'_E(1) = 9 \binom{n}{4} - cr(S) | $$ | | These relationships establish a direct connection between the coefficients of the polynomials and the crossing number, allowing for its computation in terms of algebraic quantities. | | * **Roots of the Polynomials**: The roots of the **circle polynomial** $p_C(z)$ and **Voronoi polynomial** $p_V(z)$ are critical in determining the geometric arrangement of the points. In particular, the roots of the Voronoi polynomial lie on the **unit circle** if and only if the set $S$ is in **convex position**. This result is significant because it provides a geometric characterization of the point set's configuration, linking the crossing number to the distribution of roots. | | * **Lower Bound on Root Modulus**: For sets of points in general position, bounds on the modulus of the largest root of the **circle polynomial** $p_C(z)$ and **Voronoi polynomial** $p_V(z)$ are derived. Specifically, for point sets that minimize the rectilinear crossing number $cr(S) = \alpha \cdot \binom{n}{4}$, the largest root of $p_V(z)$ is bounded below by: | | $$ | |z_{\text{max}}| \geq 1 + \frac{(1-\alpha) \pi}{16(n-3)^2} + O\left(\frac{1}{n^4}\right) | $$ | | This provides a lower bound on the root locations, which is essential for understanding the behavior of the crossing number with respect to point set configurations. | | * **Generalization to $(\leq k)$-Edge Polynomial**: A similar approach is applied to the **$(\leq k)$-edge polynomial**, yielding bounds on the largest modulus of its roots. For point sets with crossing number $cr(S) = \alpha \cdot \binom{n}{4}$, the largest root of the $p_E(z)$ polynomial is bounded below by: | | $$ | |z_{\text{max}}| \geq \frac{3 + \alpha}{9 - \alpha} | $$ | | These bounds are key in understanding the intersection between algebraic properties (roots of polynomials) and geometric configurations (the structure of the point set). | | --- | | ### **Conclusion**: | | The three polynomials—**Voronoi polynomial**, **circle polynomial**, and **$(\leq k)$-edge polynomial**—are deeply connected to the **rectilinear crossing number** of point sets. By analyzing their coefficients, derivatives, and roots, one can derive exact formulas for the crossing number and establish geometric conditions for when the roots lie on the unit circle. The results also provide bounds on the modulus of the largest roots, which are crucial for understanding the behavior of crossing numbers in various geometric configurations. | |

Checklist: "1. Model setup: Does the answer correctly define the Voronoi, circle, and (≤k)-edge polynomials and their relation to point set geometry?", | "2. Derivative formulas: Does the answer provide the derivative identities at z=1 that link these polynomials to cr(S)?", | "3. Root-based relations: Does the answer explain how cr(S) can also be expressed through sums involving the roots of the polynomials?", | "4. Unit circle characterization: Does the answer state that all roots of the Voronoi polynomial lie on the unit circle iff the point set is convex?", | "5. Bounds for non-convex sets: Does the answer provide lower bounds for the modulus of roots when the set is not convex, linked to α = cr(S)/C(n,4)?", | "6. Extension to (≤k)-edge polynomial: Does the answer include results like |z_max| ≥ (3+α)/(9-α) for its roots?", | "7. Methodological tools: Does the answer mention relevant tools such as Aziz–Mohammad identity, palindromic polynomial root theorems, Laguerre and Obrechkoff theorems, Eneström–Kakeya, and analytic bounds?", | "8. Practical implications: Does the answer connect algebraic results to geometric implications for extremal crossing numbers?" | ]