## 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.

{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 language | English |
---|---|

Pages (from-to) | 89-97 |

Number of pages | 9 |

Journal | Computational Methods and Function Theory |

Volume | 18 |

Issue number | 1 |

Early online date | 9 Oct 2017 |

DOIs | |

Publication status | Published - Mar 2018 |

## Keywords

- Root set
- Littlewood polynomials
- Unimodular polynomials