On the Location of Pseudozeros of a Complex Interval Polynomial

Given a univariate complex interval polynomial F, we provide a rigorous method for deciding whether there exists a pseudozero of F in a prescribed closed complex domain D. Here a pseudozero of F is defined to be a zero of some polynomial in F. We use circular intervals and assume that the boundary C of D is a simple curve and that C is the union of a finite number of arcs, each of which is represented by a rational function. When D is not bounded, we assume further that all the polynomials in F are of the same degree. Examples of such domains are the outside of an open disk and a half-plane with boundary. Our decision method uses the representation of C and the property that a polynomial in F is of degree 1 with respect to each coefficient regarded as a variable.      

On maximal ranges of polynomial spaces in the unit disk

Let be a domain in C, 0, and let _n ⁰ () be the set of polynomials of degreen such thatP(0)=0 andP(D), whereD denotes the unit disk. The maximal range _n is then defined to be the union of all setsP(D),P _n ⁰ (). We derive necessary and, in the case of ft convex, sufficient conditions for extremal polynomials, namely those boundaries whose ranges meet _n . As an application we solve explicitly the cases where is a half-plane or a strip-domain. This also implies a number of new inequalities, for instance, for polynomials with positive real part inD. All essential extremal polynomials found so far in the convex cases are univalent inD. This leads to the formulation of a problem. It should be mentioned that the general theory developed in this paper also works for other than polynomial spaces.Communicated by J. Milne Anderson.     

Robust Hurwitz stability via sign-definite decomposition

In this paper, we first consider the problem of determining the robust positivity of a real function f(x) as the real vector x varies over a box X∈R^l. We show that, it is sufficient to check a finite number of specially constructed points. This is accomplished by using some results on sign-definite decomposition. We then apply this result to determine the robust Hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest. We develop an eight polynomial vertex stability test that is a sufficient condition of Hurwitz stability of the family. This test reduces to Kharitonov’s well known result for the special case where the parameters are just the polynomial coefficients. In this case, the result is tight. This test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set. The result is illustrated by examples.     

Erdös-Turán theorems on a system of Jordan curves and arcs

Erdös and Turán established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [–1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadec's theorem) are given.Communicated by Dieter Gaier.     

Strong Version of the Basic Deciding Algorithm for the Existential Theory of Real Fields

Let U be a real algebraic variety in the n-dimensional affine space that is a set of all zeros of a family of polynomials of degree less than d. In the case where U is bounded (this is the main case), an algorithm of polynomial complexity is described for constructing a subset of U with the number of elements bounded from above by dⁿ that has the following property: for every s, this set has a nonempty intersection with every d-dimensional cycle with coefficients from s of the closure of the set of smooth points of dimension s of U. Bibliography: 16 titles.     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

Weighted Polynomial Approximation in the Complex Plane

Given a pair (G,W) of an open bounded set G in the complex plane and a weight function W(z) which is analytic and different from zero in G , we consider the problem of the locally uniform approximation of any function f(z) , which is analytic in G , by weighted polynomials of the form {W ⁿ (z)P _n (z) } ^$\infinity$ _n=0 , where deg P_n n. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations. May 1, 1996. Date revised: October 8, 1996.     

On arrangements of roots for a real hyperbolic polynomial and its derivatives

In this paper we count the number ?_n^(0,k), k?n−1, of connected components in the space Δ_n^(0,k) of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p(x) together with all its nonvanishing derivatives under the assumption that the roots of p(x) are real. Already the first nontrivial case n=4 shows that the obvious restrictions coming from the standard Rolle's theorem are insufficient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Base tensorielle des matrices de Hankel (ou de Toeplitz) Applications

Summary The set of all Hankel (or Toeplitz) matrices of dimensionn, is shown to possess tensorial bases: bases made ofn rank one matrices. Four families of such tensorial bases are possible. From this result, we deduce that the following computations can be performed with a number of multiplications of ordern instead of ordern ²: evaluation of the 2n+1 coefficients of the polynomial product of two polynomials of degreen, evaluation of the inverse of a lower triangular toeplitz matrix, evaluation of the quotient and of the remainder in the division of a polynomial of degree 2n by a polynomial of degreen.     

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivison Δ of a region in k ⁿ (k algebraically closed) and r∈N, there is a reflexive sheaf ? on P ⁿ , such that H ⁰(?(d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ℰ on P ⁿ in terms of the top two Chern classes and the generic splitting type of ℰ. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which ? is actually a bundle (which is always the case for n= 2). In this situation we can obtain a formula for H ⁰(?(d)) which involves only local data; the results of [9] cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P ² and we discuss a possible connection between semi-stability and the conjectured answer to this open problem. Received: 9 April 2001     

An algebra of differential operators and generating functions on the set of univalent functions

With a method close to that of Kirillov [4], we define sequences of vector fields on the set of univalent functions and we construct systems of partial differential equations which have the sequence of the Faber polynomials (F_n) as a solution. Through the Faber polynomials and Grunsky coefficients, we obtain the generating functions for some of the sequences of vector fields.     

Chebyshev polynomials for disjoint compact sets

We are concerned with the problem of finding among all polynomials of degreen with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for alln. A closely related approximation problem is obtained by considering all polynomials that have degree no larger thann and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for alln.Communicated by Doron S. Lubinsky     

The conformal ‘bratwurst’ mapsand associated Faber polynomials

Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of ‘bratwurst’ shape sets in the complex plane are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A ‘bratwurst’ shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis onto the unit circle. Such sets are well suited to serve as inclusion sets for sets associated with a matrix, for example the spectrum, field of values or a pseudospectrum. Hence, the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems. The main advantage of the approach is that the conformal maps are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic convergence factor and Faber polynomials. Numerical examples are given. Received October 7, 1998 / Revised version received March 15, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities

Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L ² for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains

   Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P _n , n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ϕ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P _n 's, the case when ϕ has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P _n 's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function.     

The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)^α(1+x)^βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.     

Explicit generalized zolotarev polynomials with complex coefficients

We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].     

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]

Letw be a suitable weight function,B _n,p denote the polynomial of best approximation to a functionf inL _w ^p [–1, 1],v _n be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB _n+1,p–B _n,p and be the arcsine measure defined by . We estimate the rate at which the sequencev _n converges to in the weak-^* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL _w ^p norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years.