The Five-Point Case of Mordell's Discriminant Inequality
Abstract
Let z1,…,z5∈ℂ satisfy ∑j=15|zj|2=5. We prove the sharp inequality ∏1≤ i<j≤5|zi−zj|2≤ 55, with equality precisely for a regular pentagon on the unit circle, up to rotation and permutation. This settles the remaining five-point case of the discriminant problem considered by Mordell in 1960. The proof first identifies a maximizer with a normalized relative equilibrium of five identical planar point vortices. We then give an independent exact classification of the corresponding stationary equations. The main algebraic step proves that every collision-free physical solution of the normalized stationary equations is reflection symmetric; it is obtained from a resultant construction and exact Gröbner-basis elimination over ℚ. The reflection-symmetric equations have exactly five physical shapes, and an exact discriminant comparison selects the regular pentagon. All symbolic eliminations, Sturm counts, and interval inequalities are reproduced by an accompanying verification script using rational arithmetic. As consequences, the same configuration gives the Gaussian weighted Fekete set and the global mode of the five-particle complex Ginibre density after the natural rescaling.
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BibTeX
@misc{wang2026fivepoint,
title = {{The Five-Point Case of Mordell's Discriminant Inequality}},
author = {Jie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00358},
eprint = {2610.00358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00358},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Five-Point Case of Mordell's Discriminant Inequality
AU - Jie Wang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00358
UR - https://arxiv.org/abs/2610.00358
AB - Let z_1,…,z_5∈ℂ satisfy ∑_j=1^5|z_j|^2=5. We prove the sharp inequality ∏_1≤ i<j≤5|z_i−z_j|^2≤ 5^5, with equality precisely for a regular pentagon on the unit circle, up to rotation and permutation. This settles the remaining five-point case of the discriminant problem considered by Mordell in 1960. The proof first identifies a maximizer with a normalized relative equilibrium of five identical planar point vortices. We then give an independent exact classification of the corresponding stationary equations. The main algebraic step proves that every collision-free physical solution of the normalized stationary equations is reflection symmetric; it is obtained from a resultant construction and exact Gröbner-basis elimination over ℚ. The reflection-symmetric equations have exactly five physical shapes, and an exact discriminant comparison selects the regular pentagon. All symbolic eliminations, Sturm counts, and interval inequalities are reproduced by an accompanying verification script using rational arithmetic. As consequences, the same configuration gives the Gaussian weighted Fekete set and the global mode of the five-particle complex Ginibre density after the natural rescaling.
ER -