Optimal Configurations for Mordell's Problem

Mordell's problem asks for the largest product of squared pairwise distances among $n$ points in the complex plane with a prescribed sum of squared distances from the origin. When this sum equals $n$, the maximum for $2\le n\le5$ is known to be $n^n$, attained precisely by the $n$th roots of unity up to rotation and relabeling. For $n\ge6$, Chalk showed that a regular polygon together with its center gives a larger product than the regular $n$-gon. We prove that for $6\le n\le9$ the maximum is $(n-1)^{n-1}r_n^{n(n-1)}$, where $r_n=\sqrt{n/(n-1)}$. Up to the same symmetries, the unique maximizer consists of the origin and the vertices of a regular $(n-1)$-gon of radius $r_n$. For six points, algebraic elimination reduces the proof to an exact comparison of stationary values. For seven points, we compare all twelve classes in a published classification of equilibria. For eight and nine points, rigorous interval arithmetic establishes an inequality for every admissible pair. Summing over pairs gives the sharp global bound, and the equality conditions determine the configuration. An explicit ten-point comparison and a known instability result together exclude Chalk's configuration for every $n\ge10$. For each $n$ from two through eight, Mordell's problem and the minimum-diameter problem studied by Bateman and Erdős have a common optimal configuration.