两个绝对有界的行列式比值
Two absolutely bounded determinantal ratios
浏览论文内容
中文总结 AI 辅助
本文针对Hall和Johnson提出的4×4正定矩阵主子式乘积比值的两个猜想，证实了R₂和R₃的上确界为1，补充了绝对有界行列式比值的相关研究。
中文摘要 AI 辅助
正定矩阵子式乘积的有界比值研究历史悠久，可追溯至1893年的Hadamard不等式。该不等式表明，对任意半正定矩阵$A$，有$$ \det A \le A_{11} \cdots A_{nn}. $$ 这一不等式随后被Fisher推广，之后又被Koteljanskii进一步拓展。Koteljanskii不等式指出，对任意半正定矩阵$A$及任意指标集$α_1, α_2 \subseteq \{1,\ldots, n\}$，有$$ \det A[α_1 \cup α_2] \det A[α_1 \cap α_2] \le \det A[α_1] \det A[α_2], $$ 其中$A[α]$表示由$α$中指标确定的主子矩阵。\n在2008年仅发布于arXiv的一篇手稿中，Hall和Johnson针对4×4正定矩阵的主子式乘积比值（记为$R_i$，$i=1,2,3$，见式(2)和(3)）提出了三个猜想。他们假设$R_1$的上确界为$27/16$，而另外两个比值的上确界为$1$。这类比值被称为绝对有界的。关于$R_1$的猜想已在文献[17]中得到证实，它是目前已知的唯一一个上确界大于1的有界行列式比值。本文的研究目标是证实关于$R_2$和$R_3$的猜想。\n已知$R_i$（$i=1,2,3$）的上界无法通过反复应用Koteljanskii不等式得到。此外，Hall和Johnson已证明$R_i$（$i=1,2,3$）的上界为4。
英文摘要
Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$ \det A \le A_{11} \cdots A_{nn}. $$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets $α_1, α_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[α_1 \cup α_2] \det A[α_1 \cap α_2] \le \det A[α_1] \det A[α_2], $$ where $A[α]$ denotes the principal submatrix determined by the indexes in $α$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.
