Related papers: Nuij type pencils of hyperbolic polynomials

We study sets of univariate hyperbolic polynomials that share the same first few coefficients and show that they have a natural combinatorial description akin to that of polytopes. We define a stratification of such sets in terms of root…

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map…

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…

We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, with a fixed base $p\in\mathbb{C}[z]$. Using a homogeneous parametrization of the family, we…

This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid…

We present a method for proving that Jensen polynomials associated with functions in the $\delta$-Laguerre-P\'olya class have all real roots, and demonstrate how it can be used to construct new functions belonging to the Laguerre-P\'olya…

We give a new and completely algebraic proof of the Helton-Vinnikov Theorem stating that every hyperbolic polynomial in three variables admits a definite linear determinantal representation.

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…

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We…

We describe a new approach to certifying the global nonnegativity of multivariate polynomials by solving hyperbolic optimization problems---a class of convex optimization problems that generalize semidefinite programs. We show how to…

For any real numbers $a,\ b$, and $c$, we form the sequence of polynomials $\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence \[ P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in\mathbb{N}, \] with the initial conditions…

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion 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…

Milnor divides all bounded hyperbolic components of cubic polynomials into 4 types (A), (B), (C) and (D). In this article, we characterize the real laminations of cubic polynomials on the tame boundary of all bounded hyperbolic components…

We give a short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have…

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes…

Let $T$ be a self-adjoint operator on a complex Hilbert space $\mathcal{H}$. We give a sufficient and necessary condition for $T$ to be the pencil $\lambda P+Q$ of a pair $( P, Q)$ of projections at some point…

A new bi-parametric $su(1,1)$ algebraization of the Heun class of equations is explored. This yields additional quasi-polynomial solutions of the form $\{z^{\alpha}P_N(z): \ \alpha \in \mathbb{C}, \ N \in \mathbb{N}_0\}$ to the General Heun…

We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the…