$σ$-Stable Matrices

关于行列式和多项式的问题

Questions about determinants and polynomials

本文讨论了系数为实根多项式系数行列式的多项式的实根性的几个问题。我们还考虑了矩阵推广为全正矩阵的一些问题，即全稳定矩阵和全上矩阵。

Differentiable roots, eigenvalues, and eigenvectors

We determine the conditions for the existence of $C^p$-roots of curves of monic complex polynomials as well as for the existence of $C^p$-eigenvalues and $C^p$-eigenvectors of curves of normal complex matrices.

枚举带标号树和森林的某些多项式矩阵的全正性，I. 有根带标号树的森林

Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees

本文研究了枚举有根带标号树森林的生成多项式矩阵，证明了其系数全正性与Hankel全正性，并推广至通用有根森林多项式，利用生产矩阵和指数Riordan数组完成证明。

Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving Möbius transformations. We obtain a characterization of Stahl--Totik regularity of a GMP ma…

Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, w…

Distances of roots of classical orthogonal polynomials

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $β$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.

枚举标号树与森林的某些多项式矩阵的全正性 II：有根标号树与偏函数有向图

Total positivity of some polynomial matrices that enumerate labeled trees and forests. II. Rooted labeled trees and partial functional digraphs

本文研究了元素为 $t_{n,k} = \binom{n}{k} n^{n-k}$ 的下三角矩阵的组合模型，证明其全正性及行生成多项式序列的 Hankel 全正性，并利用生产矩阵与指数 Riordan 阵将其推广至无穷多不定元的多项式。

Quasianalytic multiparameter perturbation of polynomials and normal matrices

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$ of $C^\infty$-functions. We require that $\mathcal C$ includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy--Carleman classes, in particular, the class of real analytic functions $C^ω$. We show that there exists a local…

调和平均、随机多项式与随机阵

Harmonic mean, random polynomials and stochastic matrices

受学习理论中一个问题的启发，我们研究了一类随机矩阵的优势特征值。这与随机多项式的导数的根有关（通过在区间[0，1]中随机均匀地选取它们的根来生成，尽管我们的结果扩展到其他分布）。这反过来又要求研究上述随机变量的调和平均的统计行为，这反过来又导致我们对稳定律的收敛速度收敛速率和稳定律的尾部估计的微妙问题。

Approximating real-rooted and stable polynomials, with combinatorial applications

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -δ$ for some $0<δ<1$. We show that for any $0 < ε<1$, the value of $p(1)$ is determined within relative error $ε$ by the coefficients $a_k$ with $k \leq {c \over \sqrtδ} \ln {n \over ε\sqrt{ δ}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < ε< 1$, the total number $M(G)=m_0(G)+m_1(G) + \ldots $ of matchings is determined within relative error $ε$ by the numbers $m_k(G)$ with $k \leq c \sqrtΔ \ln (v /ε)$…