Armin Rainer
We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.
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Nuij Sequences for Garding Hyperbolic Polynomials
本文基于 Kurdyka 和 Paunescu 的工作，针对 Garding 引入的双曲多项式空间，给出了保持双曲性算子的一个新结果。
The strong spectral order induces a natural partial ordering on the manifold $H_{n}$ of monic hyperbolic polynomials of degree $n$. We prove that twisted root maps associated with linear operators acting on $H_{n}$ are Gårding convex on every polynomial pencil and we characterize the class of polynomial pencils of logarithmic derivative type by means of the strong spectral order. Let $A'$ be the monoid of linear operators that preserve hyperbolicity as well as root sums. We show that any polynomial in $H_{n}$ is the global minimum of its $A'$-orbit and we conjecture a similar result for comple…
A real univariate polynomial with all roots real is called hyperbolic. By Descartes' rule of signs for hyperbolic polynomials (HPs) with all coefficients nonvanishing, a HP with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has exactly $c$ positive and $p$ negative roots. For $c=2$ and for degree $6$ HPs, we discuss the question: When the moduli of the $6$ roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its two positive roots depending on the positions of the two sign changes in the sequence of …
A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We consider HPs with distinct moduli of the roots. We ask the question when the order of the moduli of the negative roots w.r.t. the positive roots on the real positive half-line completely determines the signs of the coefficients of the polynomial. When th…
Let $ν_0(t),ν_1(t),\,\ldots\,,ν_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form \[R(z)=z+\sum\limits_{k=1}^n\frac{α_k}{z-μ_k},\] $μ_k$ are pairwise different real numbers, $α_k>0,\,1\leq{}k\leq{}n$. Then for each real $ξ$, the function $e^{ξν_0(t)}+e^{ξν_1(t)}+\,\cdots\,+e^{ξν_n(t)}$ is exponentially convex on the interval $-\infty<t<\infty$.
A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all $c+p$ roots are distinct and ordered on the positive half-axis, then at which positions can the $p$ moduli of negative roots be depending on the positions of the positive and negative signs of the co…
We prove that the roots of a definable $C^\infty$ curve of monic hyperbolic polynomials admit a definable $C^\infty$ parameterization, where `definable' refers to any fixed o-minimal structure on $(\mathbb R,+,\cdot)$. Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of $C^p$ (for $p \in \mathbb N$) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable and under an additional assumption also in the non-definable case. In par…
On Descartes' rule of signs for hyperbolic polynomials
研究所有根为实数且系数序列含两次符号变化的单变量实多项式，基于笛卡尔符号法则，在根模长互异条件下，详尽确定了两个正根模长在实正半轴上可能的位置。
Degree $6$ hyperbolic polynomials and orders of moduli
该研究针对系数非零的6次实根单变量多项式，解答了在根的模互不相同且排列于实正半轴的条件下，负根的模的位置如何由系数符号决定的问题。
Local well-posedness in the critical regularity setting for hyperbolic systems with partial diffusion
本文研究带部分扩散的可对称化双曲系统在临界Besov正则性大初值下的局部适定性，利用部分抛物性放宽初值正则要求，通过谱局部化和Gårding不等式结合能量方法证明结论，并以Navier-Stokes-Fourier方程为例验证。
Principal Branches of Inverse Trigonometric and Inverse Hyperbolic Functions
本文为三个关键平方根函数及反三角、反双曲函数开发主分支，利用复分析将定义转化为具体表达式，旨在扩大 DLMF 结果适用范围并降低用户的复变量负担。
We give a new self-contained proof of Bronshtein's theorem, that any continuous root of a $C^{n-1,1}$-family of monic hyperbolic polynomials of degree $n$ is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product we reprove the recent result of Colombini, Orrú, and Pernazza, that a $C^n$-curve of hyperbolic polynomials of degree $n$ admits a $C^1$-system of its roots.
A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. For $c=1$ and $2$, we discuss the question: When the moduli of all the roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its positive roots depending on the positions of the sign changes in the…
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations.…
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…
Continuity of the solution map for hyperbolic polynomials
研究证明了超几何多项式系数到根的解映射在特定结构下的连续性，并探讨了其在特征值和奇异值中的应用。
It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Klemeš is confirmed.
We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $Π_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all roots real. The subset $E_d\subset Π_d$ is the one on which a modulus of a negative root of $Q_d$ is equal to a positive root of $Q_d$. At a point, where $Q_d$ has $d$ distinct roots with exactly $s$ ($1\leq s\leq [d/2]$) equalities between positive roots and moduli of negative roots, the set $E_d$ is locally the transversal intersection of $s$ smooth hypersurface…
We study continuity of the roots of nonmonic polynomials as a function of their coefficients using only the most elementary results from an introductory course in real analysis and the theory of single variable polynomials. Our approach gives both qualitative and quantitative results in the case that the degree of the unperturbed polynomial can change under a perturbation of its coefficients, a case that naturally occurs, for instance, in stability theory of polynomials, singular perturbation theory, or in the perturbation theory for generalized eigenvalue problems. An application of our resul…
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