Choosing roots of polynomials smoothly

Choosing roots of polynomials smoothly, II

We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).

Choosing roots of polynomials with symmetries smoothly

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not $C^{1,α}$ for any $α> 0$. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials of degree $n$ admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptio…

多项式根的一种熵概念

A notion of entropy on the roots of polynomials

该研究提出了一种类比概率论随机变量熵的多项式正则熵概念，证明其随有限自由加法平滑增长，并推导得到关于多项式及其导数、非零实数的新熵不等式。

Perturbation of complex polynomials and normal operators

We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any $t_0$ there exists a positive integer $N$ such that $t \mapsto P(t_0\pm…

Optimal Sobolev regularity of roots of polynomials

We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is locally absolutely continuous with locally $p$-integrable derivatives for every $1 \le p < n/(n-1)$, uniformly with respect to the coefficients. This result is optimal: in general, the derivatives of the roots of a smooth curve of monic polynomials of degree $n$ are not locally $n/(n-1)$-integrable, and the roots may have locally unbounded variation if the coefficient…

Pólya–Schur 问题与自由概率

Pólya--Schur problems and free probability

本文在 Pólya–Schur 纲领与 Voiculescu 自由概率理论之间建立桥梁，证明任何自由（加法）无穷可分分布可作为 Appell 多项式根分布的弱极限，并推广到自由乘法无穷可分分布、矩形自由卷积及一般实根多项式，从而将热流与自由布朗运动的联系推广到任意自由 Lévy 过程。