AI 中文总结
本文证实Hall和Johnson提出的4阶正定矩阵的某行列式比的上确界为27/16，并构造出趋近于该值的矩阵序列。
AI 中文摘要
正定矩阵子式乘积的有界行列式比研究由来已久，始于将行列式上界限定为其对角元乘积的Hadamard不等式，后续发展出Fischer不等式、Koteljanskii不等式等推广形式。寻找新的有界行列式比是一项困难任务，确定其上确界更具挑战性。2008年，Hall和Johnson证明了如下比值：$$ \frac{\det A[\{1,2,4\}] \det A[\{1,3,4\}] \det A[\{2,3\}] \det A[\{1\}] \det A[\{4\}]}{\det A[\{1,2\}] \det A[ \{1,3\}] \det A[\{1,4\}] \det A[\{2,4\}] \det A[\{3,4\}]} $$ 对于所有4阶正定矩阵$A$，其上界为4，并猜想该比值的上确界为$27/16$，其中$A[\alpha]$表示$A$对应于下标集$\alpha \subseteq \{1,2,3,4\}$的主子式。本文证实该比值的上确界为$27/16$，并构造出趋近于该上确界的矩阵序列。
英文摘要
Bounded ratios between products of minors of a positive definite matrix have a long history. Starting with Hadamard's inequality which bounds from above the determinant by the product of its diagonal entries and progressing through its generalizations the Fischer's inequality and the Koteljanskii's inequality. Finding new bounded determinantal ratios is a difficult task and finding their supremum is even more difficult. In 2008, Hall and Johnson showed that the ratio $$ \frac{\det A[\{1,2,4\}] \det A[\{1,3,4\}] \det A[\{2,3\}] \det A[\{1\}] \det A[\{4\}]}{\det A[\{1,2\}] \det A[ \{1,3\}] \det A[\{1,4\}] \det A[\{2,4\}] \det A[\{3,4\}]} $$ is bounded above by $4$. They conjectured that the supremum of the ratio, over all $4 \times 4$ positive definite matrices $A$, is $27/16$. Here, $A[α]$ denotes the principal minor of $A$ corresponding to the rows and columns indexed by $α\subseteq \{1,2,3,4\}$. In this paper we confirm that the supremum of the ratio is $27/16$ and exhibit a sequence of matrices that approaches it.
