Root distribution of polynomials and distance sums on the unit circle
Kim, Eunmi
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https://hdl.handle.net/2142/44453
Description
Title
Root distribution of polynomials and distance sums on the unit circle
Author(s)
Kim, Eunmi
Issue Date
2013-05-24T22:16:39Z
Director of Research (if dissertation) or Advisor (if thesis)
Stolarsky, Kenneth B.
Doctoral Committee Chair(s)
Hildebrand, A.J.
Committee Member(s)
Stolarsky, Kenneth B.
Berndt, Bruce C.
Zaharescu, Alexandru
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2013-05-24T22:16:39Z
Keyword(s)
self-inversive polynomials
distance sum
unit circle
Abstract
Our first topic is the study of self-inversive polynomials. We establish sufficient conditions for self-inversive polynomials to have all zeros on the unit circle. We examine how such polynomials are used to generate further such polynomials. We also analyze the distribution of zeros of certain families of self-inversive polynomials and evaluate their discriminants.
Our second topic is the sum of distances between points on the unit circle. We consider the sums over the vertices of the regular $N$-gon with some stretching factors. We also consider the $N$ vertices of a regular $N$-gon with charges on the unit circle and obtain the maximal sum of squared distances from a point on the unit circle to the charged $N$-gon. For a certain set of charges, we study the minimal polynomial of the maximum value and its generating function.
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