The boundary of bounded polynomial Fatou components

We prove that, for a polynomial, every bounded Fatou component, with the exception of Siegel disks, has for boundary a Jordan curve. To cite this article: P. Roesch, Y. Yin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The polynomial Daugavetian index of a complex Banach space

We introduce the polynomial Daugavetian index of an infinite-dimensional complex Banach space. This index generalizes to polynomials the Daugavetian index defined for operators by M. Martín in 2003. We also present some results about the introduced index.     

Identification of a polynomial in nonsplitting boundary conditions

We prove criteria for the reconstructibility of n coefficients in nonsplitting boundary conditions of the Sturm-Liouville problem from n of its eigenvalues. We consider the corresponding applications, examples, and counterexamples.     

On bifurcation points of a complex polynomial

Let be a polynomial of degree . Assume that the set there is a sequence s.t. and is finite. We prove that the set of generalized critical values of (hence in particular the set of bifurcation points of ) has at most points. Moreover, We also compute the set effectively.

     

The riemann boundary problem with a complex orthogonal matrix

The Riemann boundary problem is studied under the assumption that the coefficient of the problem is a complex orthogonal matrix. In this case a property of the partial indices of the problem is established together with certain properties of the canonical matrices, which are then used to construct the canonical matrix of a complex orthogonal matrix of second order.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 405–416, March, 1978.The authors thank É. I. Zverovich, G. S. Litvinchuk, and I. M. Spitkovskii for helpful advice and remarks.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

The solution of contact problems using boundary element method

The formulation of contact problems is extended to the case of moving punches and to the case when the state of the systems being investigated depends on the history of the change in the external actions. The quasi-static contact problem for a moving rigid rough punch and a single linearly deformable body is considered. A new iterational process is proposed for solving contact problems, taking friction in the contact area into account, and its convergence is proved. An algorithm of the solution, based on the boundary element method, is developed. Solutions of specific problems are given and analysed. Estimates of the difference of the solutions due to the difference in the impenetrability conditions and the difference in the steps of the loading parameter are obtained.     

A parallel wilf algorithm for complex zeros of a polynomial

A global recursive bisection algorithm is described for computing the complex zeros of a polynomial. It has complexityO(n ³ p) wheren is the degree of the polynomial andp the bit precision requirement. Ifn processors are available, it can be realized in parallel with complexityO(n ² p); also it can be implemented using exact arithmetic. A combined Wilf-Hansen algorithm is suggested for reduction in complexity.     

The distribution of extreme points in best complex polynomial approximation

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep _n ^* (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencen _k with the property that the sequence of (n _k +2)-point Fekete subsets of \(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

Algebraic computation of the number of zeros of a complex polynomial in the open unit disk by a polynomial representation

We present a general method for the exact computation of the number of zeros of a complex polynomial inside the unit disk, assuming that the polynomial does not vanish on the unit circle. We prove the existence of a polynomial sequence. This sequence involves a reduced number of arithmetic operations and the growth of intermediate coefficients remains controlled. We study the singular case where the constant term of a polynomial of this sequence vanishes.     

Approximation methods for complex polynomial optimization

Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models where the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) symmetric conjugate form. On the constraint side, the decision variables belong to one of the following three sets: (1) the \(m\) -th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogenous polynomial functions and establish key linkages between complex multilinear forms and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented. Numerical results are reported to illustrate the effectiveness of the proposed approximation algorithms.     

Milnor number at infinity, topology and Newton boundary of a polynomial function

Abstract. In this paper we show that the Euler characteristic of the generic fibre of a complex polynomial function can be easily computed using the Newton number of f. We apply this result to study polynomials with a finite number of critical points. Received May 25, 1998; in final form January 28, 1999