通过边界刚性证明迪特尔特（Dittert）猜想
A Proof of the Dittert Conjecture via Boundary Rigidity
- Key Laboratory for Information Science of Electromagnetic Waves, College of Future Information and Technology, Fudan University（复旦大学未来信息与技术学院电磁波信息科学实验室）
- Department of Artificial Intelligence, School of Engineering, Westlake University（西湖大学工程学院人工智能系）
- University of Glasgow（格拉斯哥大学）
- School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过边界刚性方法，结合卡罗需-库恩-塔克恒等式、伯恩斯坦型正性证明等，证明了所有$n\boldsymbol{\u2265}2$时的迪特尔特猜想，还给出了部分维度的独立验证。
AI中文摘要：
设$\boldsymbol{\textit{K}}_n$为非负$n\times n$矩阵构成的单纯形，其所有元素之和为$n$。对于$\boldsymbol{\textit{K}}_n$中的矩阵$A$，记其行和为$r_1,\boldsymbol{\textit{…}},r_n$，列和为$c_1,\boldsymbol{\textit{…}},c_n$，定义$\boldsymbol{\textit{\u03a6}}(A)=\boldsymbol{\textit{\u220f}}_{i=1}^n r_i+\boldsymbol{\textit{\u220f}}_{j=1}^n c_j-\text{per}(A)$。迪特尔特猜想$\boldsymbol{\textit{\u03a6}}$的唯一最大值由均匀矩阵$U_n=J_n/n$取得，最大值为$2-n!/n^n$。我们证明了所有$n\boldsymbol{\u2265}2$时该猜想成立。黄（Hwang）定理已确定所有正的最大值点，因此核心问题是排除$\boldsymbol{\textit{K}}_n$边界上的最大值点。Cheon和Wanless的归约方法给出了不同行和不同列中的两个零元素。对于阶数为4的情况，我们对所有容许的支撑层进行分类，并通过精确的卡罗需-库恩-塔克（Karush-Kuhn-Tucker）恒等式结合伯恩斯坦（Bernstein）型正性证明排除这些层。对于所有$n\boldsymbol{\u2265}5$，一个已得层平均论证将假设的边界最大值点压缩为一个含一个重数$m=n-2$的七参数矩阵。符号奇变量展开、两个缺失元素的余子式恒等式以及均匀系数不等式随后与单侧乘子条件矛盾。该论证对$m$是统一的，且不使用有限枚举。我们还在7至13维给出了独立的谱证明，在14和15维给出了预设零证明。所有依赖维度的计算均在有理数上得到验证，并明确记录在附录中。
英文摘要：
Let $\mathcal{K}_n$ be the simplex of nonnegative $n \times n$ matrices whose entries sum to $n$. For $A$ in $\mathcal{K}_n$, with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$, define $Φ(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\mathrm{per}(A)$. Dittert conjectured that $Φ$ is uniquely maximized by the uniform matrix $U_n=J_n/n$, with maximum $2-n!/n^n$. We prove the conjecture for every $n \geq 2$. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of $\mathcal{K}_n$. A reduction of Cheon and Wanless supplies two zero entries in distinct rows and columns. In order four, we classify all admissible support strata and exclude them by exact Karush-Kuhn-Tucker identities together with a Bernstein-form positivity certificate. For all $n \geq 5$, an attained-stratum averaging argument compresses a hypothetical boundary maximizer to a seven-parameter matrix with one multiplicity $m=n-2$. A symbolic odd-variable expansion, a cofactor identity at the two missing entries, and uniform coefficient inequalities then contradict the one-sided multiplier conditions. This argument is uniform in $m$ and uses no finite enumeration. We also give independent spectral proofs in dimensions 7 through 13 and prescribed-zero proofs in dimensions 14 and 15. Every dimension-dependent calculation is certified over the rationals and recorded explicitly in the appendices.
