Hyperbolic Polynomials Research Articles

Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given Q has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for Q to be overdamped. For weakly overdamped Q the iteration is shown to be generically linearly convergent with constant at worst $1/2$, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar $\mu<0$ is provided that lies in the gap between the n largest and n smallest eigenvalues of the $n \times n$ quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$. Once such a $\mu$ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic Q so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

Remarks on the Lax conjecture for hyperbolic polynomials

A conjecture of Lax [P. Lax, Differential equations, difference equations and matrix theory, Commun. Pure Appl. Math. 11 (1958) 175–194] says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix. The conjecture has recently been proved by Lewis–Parillo–Ramana, based on previous work of Dubrovin and Helton–Vinnikov. In this note we prove related results for polynomials in several space variables which have rotational symmetries.

Hyperbolic polynomial diffeomorphisms of [formula omitted]. II: Hubbard trees

This paper is a sequel to Part I [Y. Ishii, Hyperbolic polynomial diffeomorphisms of C 2 . I: A non-planar map, Adv. Math. 218 (2) (2008) 417–464]. In the current article we construct an object analogous to a Hubbard tree consisting of a pair of trees decorated with loops and a pair of maps between them for a hyperbolic polynomial diffeomorphism f of C 2 . Key notions in the construction are the pinching disks and the pinching locus which determine how local dynamical pieces are glued together to obtain a global picture. It is proved that the shift map on the orbit space of a Hubbard tree is topologically conjugate to f on its Julia set. Several examples of Hubbard trees are also given.

Rates of recurrence for real extensions of complex dynamics

In this note, we consider the family of hyperbolic quadratic polynomials Pc (z)=z 2 + c ; c ε M, the Mandelbrot set and study ideas on the distribution of values of periodic orbits to the corresponding hyperbolic Julia sets and derive asymptotic results. Here, we alternate between symbolic dynamics and the hyperbolic polynomial map mentioned above. Later, we prove the same results for pairs of points in the Julia set. These results complement the central limit type results due to Lalley. Our proof requires application of ideas from thermodynamic formalism.

On a class of hyperbolic polynomials

In the present paper, we investigate whether the roots of a biquadratic equation determined by a pair of real symmetric positive definite matrices of order 3 and a three-dimensional vector of parameters are real. We obtain the explicit representation of the discriminant of such a polynomial as the sum of at most two squares.

Lower Bounds for Partial Matchings in Regular Bipartite Graphs and Applications to the Monomer–Dimer Entropy

We derive here the Friedland–Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three-dimensional monomer–dimer entropy.

Holomorphic motions and structural stability of polynomial automorphisms of \mathbf{C}^{2}

Combining ideas from real dynamics on compact manifolds and complex dynamics in one variable, we prove the structural stability of hyperbolic polynomial automorphisms in C 2 . We consider families of hyperbolic polynomial automorphisms depending holomorphically on the parameter A. This is done over a series of steps - given a family {f λ }, where |λ| is sufficiently small, we construct mappings defined on a neighborhood U of J 0 which conjugate f 0 and f λ . Moreover, it is shown that J moves holomorphically. This conjugacy is then used to construct a conjugacy between f 0 and f λ defined on a neighborhood M of J + 0 ∪ J - 0 . Finally, we extend such a mapping to construct a conjugacy on all of C 2 .

Distances between critical points and midpoints of zeros of hyperbolic polynomials

Let p ( x ) be a polynomial of degree n with only real zeros x 1 ⩽ x 2 ⩽ ⋯ ⩽ x n . Consider their midpoints z k = ( x k + x k + 1 ) / 2 and the zeros ξ 1 ⩽ ξ 2 ⩽ ⋯ ⩽ ξ n − 1 of p ′ ( z ) . Motivated by a question posed by D. Farmer and R. Rhoades, we compare the smallest and largest distances between consecutive ξ k to the ones between consecutive z k . The corresponding problem for zeros and critical points of entire functions of order one from the Laguerre–Pólya class is also discussed.

The Schur–Szegö composition for hyperbolic polynomials

The composition of Schur–Szegö of the polynomials P ( x ) = ∑ j = 0 n C n j a j x j and Q ( x ) = ∑ j = 0 n C n j b j x j is defined as P ∗ Q = ∑ j = 0 n C n j a j b j x j . In the case when P and Q are hyperbolic, i.e. with real roots only, we give the exhaustive answer to the question if the numbers of positive, negative and zero roots of P and Q are known what these numbers can be for P ∗ Q . To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

On multiplier sequences

In the present paper we consider real polynomials in one real variable of a given degree n . Such a polynomial is called hyperbolic if it has only real roots. A finite multiplier sequence of length n + 1 (FMS( n + 1 )) is a tuple ( c 0 , … , c n ) , c i ∈ R , such that if ∑ i = 0 n b i x n − i , b i ∈ R , is a hyperbolic polynomial, then ∑ i = 0 n c i b i x n − i is also such a polynomial. The set of FMS( n + 1 ) coincides with the set of tuples such that ∑ i = 0 n C n i c i x n − i is a hyperbolic polynomial with all roots of the same sign. In the paper we prove several geometric properties of the set of FMS( n + 1 ) formulated in terms of its stratification (defined by the multiplicity vectors of the polynomials) and of the Whitney property (the curvilinear distance to be equivalent to the Euclidean one).

Choosing Roots of Polynomials with Symmetries Smoothly

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C1,a for any a > 0. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a C3n curve of hyperbolic polynomials of degree n admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.

On hyperbolic polynomial-like functions and their derivatives

A real polynomial $P$ of degree $n$ in one real variable is hyperbolic if its roots are all real. A real-valued function $P$ is called a hyperbolic polynomial-like function of degree $n$ if it has $n$ real zeros and $P^{(n)}$ vanishes nowhere. Denote by $x_k^{(i)}$ the roots of $P^{(i)}$ , $k=1,\dots,n-i$ , $i=0,\dots,n-1$ . Then, in the absence of any equality of the form \begin{equation} x_i^{(j)}=x_k^{(l)} \tag{*} \label{*} \end{equation} for all $i we have \begin{equation} x_k^{(i)}<x_k^{(j)}<x_{k+j-i}^{(i)} \tag{**} \label{**} \end{equation} (the Rolle theorem). For $n\geq4$ (respectively, for $n\geq5$ ) not all arrangements without equalities \eqref{*} of $\tfrac12n(n+1)$ real numbers $x_k^{(i)}$ and compatible with \eqref{**} are realizable by the roots of hyperbolic polynomials (respectively, of hyperbolic polynomial-like functions) of degree $n$ and of their derivatives. For $n=5$ and when $$ x_1^{(1)}<x_1^{(3)}<x_2^{(1)}<x_3^{(1)}< x_2^{(3)}<x_4^{(1)} $$ we show that all such 102 arrangements are realizable by hyperbolic polynomial-like functions (of which 66 are obtained by hyperbolic polynomials and another 8 by perturbations of such).

A counting result on two-dimension with error terms

In this article, we study ideas on the distribution of values of ergodic sums of functions on periodic orbits of product hyperbolic quadratic polynomials defined on their product Julia sets. Towards this end, we use ideas from thermodynamic formalism and basic Fourier transforms. This generalises a result due to the author [in Sridharan, S., Asymptotic results for hyperbolic quadratic polynomials (Preprint)] in the two-dimensional case and we obtain an error term for functions that satisfy a diophantine condition.

A general class of Bernstein-like bases

This paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials. Bases with optimal shape preserving and stability properties are constructed. Evaluation and subdivision algorithms for them are provided. Bases for the corresponding and mixed spline spaces are also constructed. Some nice properties of these bases and the generated curves are shown.

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable \({\phi^{4}_4}\) quantum field theory on the Moyal non commutative \({\mathbb R^{4}}\) space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information.

Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron–Frobenius–Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient conditions for two-weight boundedness of the Hilbert transform. It has been proved in [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2–3) (2006) 215–230] for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t = 0 that the Lipschitz estimate becomes exponentially better when the dimension of the Jacobi matrix grows. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. We suggest a scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the present paper and those of [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2–3) (2006) 215–230]. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron–Frobenius–Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron–Frobenius–Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.

An orthogonal basis for the hyperbolic hybrid polynomial space

Motivated by the wide usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bezier basis in the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bezier basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the H- Bezier curves is sketched to illustrate the utility of the orthogonal basis.

On root arrangements for hyperbolic polynomial-like functions and their derivatives

A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function ( HPLF) of degree n if it has n real zeros and P ( n ) vanishes nowhere. Denote by x k ( i ) the roots of P ( i ) , k = 1 , … , n − i , i = 0 , … , n − 1 . Then in the absence of any equality of the form x i ( j ) = x k ( l ) ( ∗ ) one has ∀ i < j , x k ( i ) < x k ( j ) < x k + j − i ( i ) ( ∗ ∗ ) (the Rolle theorem). For n ⩾ 4 (resp. for n ⩾ 5 ) not all arrangements without equalities ( ∗ ) of n ( n + 1 ) / 2 real numbers x k ( i ) and compatible with ( ∗ ∗ ) (we call them admissible) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and of their derivatives. For n = 5 we show that from 286 admissible arrangements, exactly 236 are realizable by HPLFs; from these 236 arrangements, 116 are realizable by hyperbolic polynomials and 24 by perturbations of such.

Nonsmooth Analysis of Lorentz Invariant Functions

A real valued function $g(x,t)$ on ${\mathbb{R}}^n \times {\mathbb{R}}$ is called a Lorentz invariant if $g(x,t)=g(Ux,t)$ for all $n \times n$ orthogonal matrices U and all $(x,t)$ in the domain of g. In other words, g is invariant under the linear orthogonal transformations preserving the Lorentz cone: $\{(x,t) \in {\mathbb{R}}^n \times {\mathbb{R}} \,|\, t \ge \|x\| \}$. It is easy to see that every Lorentz invariant function can be decomposed as $g=f \circ \beta$, where $f : {\mathbb{R}}^2 \rightarrow {\mathbb{R}}$ is a symmetric function and $\beta$ is the root map of the hyperbolic polynomial $p(x,t)=t^2-x_1^2-\cdots -x_n^2$. We investigate a variety of important variational and nonsmooth properties of g and characterize them in terms of the symmetric function f.

Convexity Properties of Twisted Root Maps

The strong spectral order induces a natural partial ordering on the manifold Hn of monic hyperbolic polynomials of degree n. We prove that twisted root maps associated with linear operators acting on Hn are Gu arding convex on every polynomial pencil and we characterize the class of polynomial pencils of logarithmic derivative type by means of the strong spectral order. Let Abe the monoid of linear operators that preserve hyperbolicity as well as root sums. We show that any polynomial in Hn is the global minimum of its A ' -orbit and we conjecture a similar result for complex polynomials.