秩-2 永久式与有限自由卷积的不等式
Inequalities for rank-two permanents and finite free convolutions
AI总结：
该研究强化了Bang的矩阵永久式不等式，将其适用范围扩展至秩不超过2的实矩阵，并利用该强化结果推导得出多项式有限自由卷积的新不等式。
AI中文摘要：
Bang（1976）证明了矩阵永久式的不等式：$\text{per}^2(A) \geq 2^{-2n}\text{per}(A \otimes J_2)$，其中$J_2$是2×2全1矩阵，$A$是任意$n$阶非负元素矩阵。本文证明，若$A$是任意秩不超过2的$n$阶实矩阵（可含负元素），该不等式可被强化，将常数$2^{-2n}$替换为$1/\binom{2n}{n} = (n!)^2/(2n)! > 2^{-2n}$。随后证明该强化不等式可导出多项式有限自由卷积的新不等式：若$p$和$q$是$n$次首一实根多项式，则对所有$x \in \mathbb{R}$，有$(p \boxplus_n q)(x)^2 \geq (p^2 \boxplus_{2n} q^2)(x)$和$(p \boxtimes_n q)(x)^2 \geq (p^2 \boxtimes_{2n} q^2)(x)$，其中$\boxplus_n$和$\boxtimes_n$分别是$n$次多项式上的有限自由加法与乘法卷积运算。
英文摘要：
Bang (1976) proved the inequality for matrix permanents $\mathrm{per}^2(A) \geq 2^{-2n}\mathrm{per}(A \otimes J_2)$, where $J_2$ is the $2 \times 2$ all-ones matrix and $A$ is any $n \times n$ matrix with non-negative entries. We show that, if $A$ is any $n \times n$ real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant $2^{-2n}$ by $1 / \binom{2n}{n} = (n!)^2 / (2n)! > 2^{-2n}$. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if $p$ and $q$ are monic real-rooted polynomials of degree $n$, then $(p \boxplus_n q)(x)^2 \geq (p^2 \boxplus_{2n} q^2)(x)$ and $(p \boxtimes_n q)(x)^2 \geq (p^2 \boxtimes_{2n} q^2)(x)$ for all $x \in \mathbb{R}$, for $\boxplus_n$ and $\boxtimes_n$ the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree $n$.
