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

Simon Baker, Han Yu

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)89-97
Number of pages9
JournalComputational Methods and Function Theory
Volume18
Issue number1
Early online date9 Oct 2017
DOIs
Publication statusPublished - Mar 2018

Keywords

  • Root set
  • Littlewood polynomials
  • Unimodular polynomials

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