单位三角形中的海尔布伦问题：\(n\leq8\)时的认证最优配置
Heilbronn's Problem in the Unit Triangle: Certified Optimal Configurations for up to $n\le 8$
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中文总结 AI 辅助
研究单位直角三角形中\(n\)个点构成三角形最小面积最大的海尔布伦问题，通过边界结构结果和仿射\(S_3\)对称性确定混合整数模型，证明\(n\leq8\)全局最优，给出\(n\leq7\)精确解，重构\(n = 8\)最优解并确定其伽罗瓦群。
中文摘要 AI 辅助
我们研究单位直角三角形中的海尔布伦三角形问题，即放置\(n\)个点以最大化它们所构成的\(\binom{n}{3}\)个三角形面积中的最小面积。我们证明了一个边界结构结果：除非三个顶点都被占据，对于\(n\geq5\)的某些最优配置在边界上至少有四个点，其中一条边上有两个点。利用仿射\(S_3\)对称性，这在一个混合整数模型中确定了四个边界点和\(n\)个方向变量，该模型证明了所有\(n\leq8\)（包括\(n = 7, 8\)，之前没有证明）的全局最优性，填补了网格搜索和分支定界法留下的空白。对于\(n\leq7\)，我们得到了具有明确配置的精确最优解。对于\(n = 8\)，最优解推测是陈、曾和周得到的一个七次方程的实根，我们的重构将其精确到\(250\)位小数。我们表明其伽罗瓦群是\(S_7\)，所以基于该推测不存在根式表达式。
英文摘要
We study Heilbronn's triangle problem in the unit right triangle, where $n$ points are placed to maximize the smallest of the $\binom{n}{3}$ triangle areas they span. We prove a boundary-structure result: unless all three vertices are occupied, some optimal configuration with $n \ge 5$ has at least four points on the boundary, one edge carrying two of them. With the affine $S_3$ symmetry this fixes four boundary points and $n$ orientation variables in a mixed-integer model that certifies global optimality for all $n \le 8$: for $n = 8$ apparently the first proof, and for $n = 7$ an independent confirmation of the symbolic-computation proof of Zeng and Chen. For $n \le 7$ we obtain exact optima with explicit configurations. For $n = 8$ the optimum is conjectured to be the real root of a septic obtained by Chen, Zeng and Zhou, which our reconstruction confirms to $250$ digits. We show its Galois group is $S_7$, so on that conjecture no expression in radicals exists.
