Hyperbolic Polynomials Research Articles

Special homogeneous surfaces

Abstract We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

NESTED POLYNOMIALS TRIGONOMETRIC AND HYPERBOLIC TYPE WITH APPLICATIONS

<p class="ABSTRACT">In this paper we introduce nested trigonometric and hyperbolic polynomials of rank n by employing nested trigonometric and hyperbolic functions, respectively. Our investigation delves into some interesting properties of nested polynomials, establishing some recurrence formulas and deriving some useful summation formulas. Additionally, we establish a connection between nested trigonometric polynomials and one specific class of special polynomials that satisfy a second-order Sturm-Liouville differential equation. Finally, we outline several applications of nested polynomials. Among others, it is shown that nested trigonometric polynomials can be used for the linear static analysis of curved Bernoulli–Euler beam.</p>

Hyperbolic and Weak Euclidean Polynomials from Wronskian and Leibniz Maps

A real univariate polynomial is called hyperbolic or stable if all its roots are real. We search for hyperbolic polynomials of two and three degrees by using the Wronskian map W and a dual map to W called Leibniz, since it involves the classical Leibniz rule for the derivative of a product of functions. In addition to hyperbolicity, we use these two methods to search for a class of polynomials introduced by the first author and now called weak Euclidean.

On Descartes' rule of signs for hyperbolic polynomials

We consider univariate real polynomials with all real roots and with two sign changes in the sequence of their coefficients which are all non-vanishing. Assume that one of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree \(d\) polynomials have 2 positive and \(d-2\) negative roots. We consider the possible sequences of the moduli of their roots on the real positive half-axis. When these moduli are distinct, we give the exhaustive answer to the question which positions can the moduli of the two positive roots occupy.

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

Comparison principles for nonlinear potential theories and PDEs with fiberegularity and sufficient monotonicity

We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets F⊂J2(X) of the 2-jets on X⊂Rn open) and subsolutions of degenerate elliptic and parabolic PDEs of the form F(x,u,Du,D2u)=0. We will implement the monotonicity-duality method begun by Harvey and Lawson (2009) (in the pure second order constant coefficient case) for proving comparison principles for potential theories where F has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators F which are compatible with F in a precise sense for which the correspondence principle holds.We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary.Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Examples operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of Gårding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold.

Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank 1, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.

The Laguerre-Pólya Class and Combinatorics

The talks at the workshop were focused on zero localization and zero finding of entire functions, with applications to analytic number theory and combinatorics. The discussions included specific areas such as stable and hyperbolic polynomials, the Laguerre-Pólya class of entire functions, Pólya frequency sequences, total positivity for sequences and functions, and zeros of generating functions arising in probability and combinatorics.

Commutativity, Majorization, and Reduction in Fan–Theobald–von Neumann Systems

A Fan–Theobald–von Neumann system (Gowda in Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ) is a triple $$({{\mathcal {V}}},{{\mathcal {W}}},\lambda )$$ , where $${{\mathcal {V}}}$$ and $${{\mathcal {W}}}$$ are real inner product spaces and $$\lambda :{{\mathcal {V}}}\rightarrow {{\mathcal {W}}}$$ is a norm-preserving map satisfying a Fan–Theobald–von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In Gowda (Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ), we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan–Theobald–von Neumann-type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan–Theobald–von Neumann systems.

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

Abstract A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.

Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

AbstractThis article is concerned with the convergence analysis of an extension of the locally optimal preconditioned conjugate gradient method for the extreme eigenvalue of a Hermitian matrix polynomial that possesses some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J Numer Anal. 2008;46(5):2567–92.). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence behavior.

On adjoint torsion polynomial of genus one two-bridge knots

Dunfield, Friedl and Jackson make a conjecture that the hyperbolic torsion polynomial determines the genus and fibering of hyperbolic knots. In this paper, we study a similar problem for the adjoint torsion polynomial, and show that it determines the genus and fibering of a large family of hyperbolic genus one two-bridge knots.

Roots of Gårding hyperbolic polynomials

We explore the regularity of the roots of Gårding 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.

Hyperbolic polynomials and rigid orders of moduli

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 there is at least one positive and at least one negative root this is possible exactly when the moduli of the negative roots interlace with the positive roots (hence half or about half of the roots are positive). In this case the signs of the coefficients of the HP are either $(+,+,-,-,+,+,-,-,\ldots )$ or $(+,-,-,+,+,-,-,+,\ldots )$.

Modeling Core Parts of Zakeri Slices I

The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them \emph{Zakeri slices}. We give a model of the central part of a slice (the subset of the slice that can be approximated by hyperbolic polynomials with Jordan curve Julia sets), and a continuous projection from the central part to the model. The projection is defined dynamically and is coherent with the dynamical-analytic parameterization of the Principal Hyperbolic Domain by Petersen and Tan Lei.

Refinements of Huygens-Wilker-Lazarovic inequalities via the hyperbolic cosine polynomials

The aim of this paper is to provide new refinements of Huygens-Wilker-Lazarovic inequalities using hyperbolic cosine polynomials. We give an unitary approach for both inequalities of trigonometric and hyperbolic functions.

On Certain Class of Weighted Hyperbolic Polynomials

On Certain Class of Weighted Hyperbolic Polynomials

A spectral characterization and an approximation scheme for the Hessian eigenvalue

We revisit the k -Hessian eigenvalue problem on a smooth and bounded {(k-1)} -convex domain in \mathbb{R}^n . First, we obtain a spectral characterization of the k -Hessian eigenvalue as the infimum of the first eigenvalues of linear second-order elliptic operators whose coefficients belong to the dual of the corresponding Gårding cone. Second, we introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the k -Hessian operator. We show that the scheme converges, with a rate, to the k -Hessian eigenvalue for all k . When 2\leq k\leq n , we also prove a local L^1 convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.

Primitive tuning via quasiconformal surgery

Using quasiconformal surgery, we prove that any primitive, postcritically-finite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable.