发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Mordell判别不等式五点情形的尖锐形式，通过点涡旋平衡与代数分类确定极大值为正五边形，并给出验证脚本。
AI 中文摘要
设 $z_1,\ldots,z_5\in\mathbb C$ 满足 $\sum_{j=1}^5|z_j|^2=5$。我们证明尖锐不等式 \\[ \prod_{1\le i<j\le5}|z_i-z_j|^2\le 5^5, \\] 等号当且仅当在单位圆上取正五边形（在旋转和置换意义下）。这解决了Mordell在1960年提出的判别式问题的剩余五点情形。证明首先确定一个极大化子与五个相同平面点涡旋的归一化相对平衡相对应。然后我们给出相应平稳方程的独立精确分类。主要代数步骤证明归一化平稳方程的每个无碰撞物理解都是反射对称的；该步骤通过结式构造和在 $\mathbb Q$ 上的精确Gröbner基消元获得。反射对称方程恰好有五个物理形状，精确判别式比较选出正五边形。所有符号消元、Sturm计数和区间不等式均由随附的验证脚本使用有理算术重现。作为推论，相同构型在自然重标度下给出高斯加权Fekete集和五粒子复Ginibre密度的全局模态。
英文摘要
Let $z_1,\ldots,z_5\in\mathbb C$ satisfy $\sum_{j=1}^5|z_j|^2=5$. We prove the sharp inequality \[ \prod_{1\le i<j\le5}|z_i-z_j|^2\le 5^5, \] 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 $\mathbb Q$. 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.
Comments18 pages
