Optimal Configurations for Mordell's Problem
Abstract
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≤ n≤5 is known to be nn, attained precisely by the nth roots of unity up to rotation and relabeling. For n≥6, Chalk showed that a regular polygon together with its center gives a larger product than the regular n-gon. We prove that for 6≤ n≤9 the maximum is (n−1)n−1rnn(n−1), where rn=√(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 rn. 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≥10. 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.
Related
- On the Five-Point Case of Mordell's Problem
- Finiteness of Dziobek central configurations in the Newtonian five-body problem
- On Grünbaum's problem for symmetric configurations
- Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem
- Uniqueness of convex central configurations in the planar (1 + 4)-body problem
BibTeX
@misc{ji2026optimal,
title = {{Optimal Configurations for Mordell's Problem}},
author = {Xiang Ji and Tiancheng Xu and Wensheng Yu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03088},
eprint = {2611.03088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03088},
}
RIS
TY - UNPB
TI - Optimal Configurations for Mordell's Problem
AU - Xiang Ji
AU - Tiancheng Xu
AU - Wensheng Yu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03088
UR - https://arxiv.org/abs/2611.03088
AB - 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≤ n≤5 is known to be n^n, attained precisely by the nth roots of unity up to rotation and relabeling. For n≥6, Chalk showed that a regular polygon together with its center gives a larger product than the regular n-gon. We prove that for 6≤ n≤9 the maximum is (n−1)^n−1r_n^n(n−1), where r_n=√(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≥10. 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.
ER -