Hyperbolic Polynomials Research Articles

On some combinatorial formulae coming from Hessian Topology

We prove that the fields of asymptotic lines of a real hyperbolic homogeneous polynomial are isotopic to the corresponding fields of its hyperbolic homogeneous part. We also show some combinatorial identities which are related to such isotopy.

Level‐set topology optimization for robust design of structures under hybrid uncertainties

SummaryThis paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen‐Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen‐Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst‐case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.

A characterization of round spheres in space forms

Let $\mathbb Q^{n+1}_c$ be the complete simply-connected $(n+1)$-dimensional space form of curvature $c$. In this paper we obtain a new characterization of geodesic spheres in $\mathbb Q^{n+1}_c$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in $\mathbb Q^{n+1}_c,\;c\leq 0,$ with constant mean curvature and constant scalar curvature. The proof relies on the well known Omori-Yau maximum principle, a formula of Walter for the Laplacian of the $r$-th mean curvature of a hypersurface in a space form, and a classical inequality of G\aa rding for hyperbolic polynomials.

Symbolic computation in hyperbolic programming

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming (SDP). We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.

Determinantal representations of invariant hyperbolic plane curves

We study hyperbolic polynomials with nice symmetry and express them as the determinant of a Hermitian matrix with special structure. The goal of this paper is to answer a question posed by Chien and Nakazato in (2015) [3]. By properly modifying a determinantal representation construction of Dixon (1902) [5], we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix. Moreover, if the polynomial is invariant under the action of the dihedral group of order n, the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries.

Non-representable hyperbolic matroids

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vámos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids.We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vámos matroid and the generalized Vámos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre–Turán inequality and an inequality due to Jensen. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

A continuity result on quadratic matings with respect to parameters of odd denominator rationals

AbstractIn this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-P\'{o}lya class via hyperbolic polynomials play an important role in this paper. Some open problems are also stated, which may be of interest for further research.

Interpolation mixing hyperbolic functions and polynomials

Exponential polynomials as solutions of differential equations with constant coefficients are widely used for approximation purposes. Recently, mixed spaces containing algebraic, trigonometric and exponential functions have been extensively considered for design purposes. The analysis of these spaces leads to constructions that can be reduced to Hermite interpolation problems. In this paper, we focus on spaces generated by algebraic polynomials, hyperbolic sine and hyperbolic cosine. We present classical interpolation formulae, such as Newton and Aitken-Neville formulae and a suggestion of implementation. We explore another technique, expressing the Hermite interpolant in terms of polynomial interpolants and derive practical error bounds for the hyperbolic interpolant.

Livsic-type determinantal representations and hyperbolicity

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X⊂Pd of an arbitrary codimension ℓ with respect to a real ℓ−1-dimensional linear subspace V⊂Pd and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces (ℓ=1), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in Pd hyperbolic with respect to some real d−2-dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.

On the connectivity of the hyperbolicity region of irreducible polynomials

AbstractWe give a proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones. Apart from the use of Bertini’s Theorem the proof is elementary.

Asymptotic Estimation and Existence of A Coupled System Model Based on Symmetric Space of Riemannian Manifold

In this paper, we will discuss the fully nonlinear elliptic and parabolic equations related to classical Euclidean geometry and conformal geometry. Some algebraic and analytic properties of concave symmetric functions and Garding’s theory of hyperbolic polynomials are collected in the appendix. According to the classification theory of Riemann symmetry space, we use transformation to convert the subalgebra of a very tight part to a very noncompact subalgebra. By calculating the projection, we calculate the section curvature of all irreducible Riemann symmetric spaces. Using the classification theory of the maximal subfamily of abstract roots and the control chart, we calculate the partial positive partial negative values of the curvature of all irreducible Riemann symmetric spaces.

Radii of starlikeness and convexity of a cross-product of Bessel functions

In this paper, some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product, three different normalizations are investigated and for each of the six functions, the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike in the open unit disk; however, the convex case is open for further research. The characterization of entire functions from the Laguerre–Polya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing properties of the zeros of the cross-product and product of Bessel functions and their derivatives are also useful in the proof of the main results.

Nuij Type Pencils of Hyperbolic Polynomials

AbstractNuij’s theorem states that if a polynomial p ∈ ℝ[z] is hyperbolic (i.e., has only real roots), then p+sp'' is also hyperbolic for any s ∈ ℝ. We study other perturbations of hyperbolic polynomials of the form pa(z, s) := . We give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which pa(z, s) is a pencil of hyperbolic polynomials. We also give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which the associated families pa(z, s) admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

On the Pochhammer transformation and hyperbolic polynomials decomposed in the Pochhammer basis

For a given we consider the linear operator on the space of complex polynomials given by the formula We call the h-Pochhammer transformation. In this paper we investigate the properties of the Pochhammer transformation, in particular, how it affects the first and last roots of a hyperbolic polynomial. We study how the coefficients in the decomposition of a polynomial with respect to the Pochhammer basis are related to its roots. We also prove an h-analogue of the famous Schur Convolution Theorem.

On the Roots of a Hyperbolic Polynomial Pencil

Let \(\nu _0(t),\nu _1(t),\ldots ,\nu _n(t)\) be the roots of the equation \(R(z)=t\), where R(z) is a rational function of the form $$\begin{aligned} R(z)=z-\sum \limits _{k=1}^n\frac{\alpha _k}{z-\mu _k}, \end{aligned}$$ \(\mu _k\) are pairwise distinct real numbers, \(\alpha _k>0,\,1\le {}k\le {}n\). Then for each real \(\xi \), the function \(e^{\xi \nu _0(t)}+e^{\xi \nu _1(t)}+\,\cdots \,+e^{\xi \nu _n(t)}\) is exponentially convex on the interval \(-\infty<t<\infty \).

Determinantal representations of semihyperbolic polynomials

We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.

Determinantal representations and Bézoutians

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms called B\'ezoutians. Besides commutative algebra, the proof relies on results from real algebraic geometry.

Offset Approximation of Hybrid Hyperbolic Polynomial Curves

In this paper, two methods for approximation of offset curves of H-Bezier curves and H-B spline curves are presented, which are based on the approximation of shifting control points and norm of parametric speed. Firstly, after calculating the perturbed vectors, the offset curve can be obtained by shifting the control points of the base curve. Then, both the Chebyshev approximation and optimal trigonometric polynomials approximation of parametric speed of base curve are presented, and two approximation functions of offset curves are also obtained. Finally, some examples are used to demonstrate their practicality.