Related papers: Continuity of the solution map for hyperbolic poly…
Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…
We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=\varphi,$ for hyperbolic Poisson's equation $\Delta_h u=\mu$ where $\varphi\in L_1(\partial \mathbb{B}^n)$ and $\mu$ is a measure that satisfies a growth condition. Next we…
We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic…
We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots.…
In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then…
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 $\mathcal C$…
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and…
We study classes of locally biholomorphic mappings defined in the $\P$ that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…
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…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
In this paper, we show that harmonic Bloch mappings are Lipschitz continuous with respect to the pseudo-hyperbolic metric. This result improves the corresponding result of Theorem 1 of [P. Ghatage, J. Yan, and D. Zheng, Composition…
We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…
We study continuous approximate solutions to polynomial equations over the ring $C(X)$ of continuous complex-valued functions over a compact Hausdorff space $X$. We show that when $X$ is one-dimensional, the existence of such approximate…
Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential…
We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there…
We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $[0,T]\times \R^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay…
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex…
Hyperbolic polynomials have been of recent interest due to applications in a wide variety of fields. We seek to better understand these polynomials in the case when they are symmetric, i.e. invariant under all permutations of variables. We…
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…