Root sets of polynomials and power series with finite choice of coefficients

The first author is supported by the EPSRC Grant EP/M001903/1. The second author is supported by a PhD scholarship provided by the School of Mathematics in the University of St Andrews.Given H⊆C two natural objects to study are the set of zeros of polynomials with coefficients in H, {z∈C:∃k>0,∃(an)∈Hk+1,∑n=0kanzn=0}, and the set of zeros of a power series with coefficients in H, {z∈C:∃(an)∈HN,∑n=0∞anzn=0}. In this paper, we consider the case where each element of H has modulus 1. The main result of this paper states that for any r∈(1/2,1), if H is 2cos−1(5−4|r|24)-dense in S1, then the set of zeros of polynomials with coefficients in H is dense in {z∈C:|z|∈[r,r−1]}, and the set of zeros of power series with coefficients in H contains the annulus {z∈C:|z|∈[r,1)}. These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them as H becomes progressively more dense.Publisher PDFPeer reviewe

Smale's mean value conjecture for finite Blaschke products

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.Comment: To appear in an issue of Journal of Analysis denoted to the Proceedings of the Conference on Modern Aspects of Complex Geometry (MindaFest)

Braids, Conformal Module and Entropy

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conformal module is inverse proportional to a popular dynamical braid invariant, the entropy. The entropy appeared in connection with Thurston's theory of surface homeomorphisms. An application of the concept of conformal module to algebraic geometry is given.Comment: Research announcement, 4 pages Minor revision, misprints correcte

Szego polynomials: a view from the Riemann-Hilbert window

This is an expanded version of the talk given at the conference ``Constructive Functions Tech-04''. We survey some recent results on canonical representation and asymptotic behavior of polynomials orthogonal on the unit circle with respect to an analytic weight. These results are obtained using the steepest descent method based on the Riemann-Hilbert characterization of these polynomials.Comment: 23 pages, 7 figures; To appear in Elect. Trans. Num. Anal. Apparently, due to some missing .sty file, all integrals in version 2 were gone. Version 3 is a mere correction of this proble