Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.3087

The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

Julius Borcea·Petter Brändén
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.

Explore related subjects

math.CVcond-mat.stat-mechmath-phmath.COmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Julius Borcea, Petter Brändén. 2008-09-18. The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications. https://doi.org/10.1002/cpa.20295

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part

In the present article, we discuss about the estimate of the pre-Schwarzian and Schwarzian norms for locally univalent harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$. First, we prove a general result for the estimate of the pre-Schwarzian norm which rectify few earlier flawed results. We also consider a new class $\mathcal{F}_0$ consisting of all harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that ${\rm Re\,}\left(1+z\frac{h''(z)}{h'(z)}\right)>0$ for $z\in\mathbb{D}$ with dilatation $ω_f(z)\in Aut(\mathbb{D})$ and obtain best possible estimates of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion and coefficient estimates of the co-analytic function $g$ when $f=h+\overline{g}\in\mathcal{F}_0$.

math.CV↗
arXiv2026

The Reciprocal Problem on Weighted Bergman Spaces

The reciprocal problem on weighted Bergman spaces has been posed as an open problem. In this paper, we establish several sufficient conditions for the reciprocal property and clarify the parameter ranges in which the available methods are applicable. In particular, we prove that functions in $A_α^p\cap H^\infty$ enjoy the reciprocal property in the parameter ranges where the required analytic Besov composition theorem is available. In addition, using Hardy boundary estimates, we solve the reciprocal problem in the Drury--Arveson space $H_d^2$ when the dimension is $d=3$, and give an equivalent condition for the reciprocal problem in the four-dimensional Drury--Arveson space.

math.CV↗
arXiv2026

Solving non-oscillatory solutions of the Hill equation via the Tumura--Clunie method

We consider the Hill equation $f''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}e^{iz})f=0$ ($†$), where $\mathbf{k}\geq 1$ and $\mathbf{l}\geq 0$ are integers and $b_{-\mathbf{l}}$, $\cdots$, $b_{\mathbf{k}}$ are constants such that $b_{\mathbf{k}}\not=0$. We point out that there is a full correspondence between the class of non-oscillatory solutions such that $λ(f)<\infty$ of equation ($†$) and the class of Liouvillian solutions of equation $x^2u''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}x^{i})u=0$ ($‡$). Then this paper has twofold purposes. First, parallel to Kovacic's algorithms to find the Liouvillian solutions of equation ($‡$), we develop the Tumura--Clunie method to find the non-oscillatory solutions of a higher order version of the Hill equation. In this part, we first determine the form of entire solutions of a general Tumura--Clunie type differential equation. Second, for the particular Hill equation $f''-(b_{\mathbf{k}}e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{k}}b_{\mathbf{s}}\not=0$, we use the Tumura--Clunie method to determine the non-oscillatory solution $f$ with an additional zero property.

math.CV↗