Random Harmonic Polynomials

Date Issued:: 2017
Summary:: The study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization. Three prevalent models of signi cant study are the Kostlan model, the Weyl model, and the naive model in which the coe cients of the polynomial are standard Gaussian random variables. A harmonic polynomial is a complex function of the form h(z) = p(z) + q(z) where p and q are complex analytic polynomials. Li and Wei adapted the Kac integral formula for the expected number of zeros to study random harmonic polynomials and take particular interest in their interpretation of the Kostlan model. In this thesis we nd asymptotic results for the number of zeros of random harmonic polynomials under both the Weyl model and the naive model as the degree of the harmonic polynomial increases. We compare the ndings to the Kostlan model as well as to the analytic analogs of each model. We end by establishing results which lead to open questions and conjectures about random harmonic polynomials. We ask and partially answer the question, \When does the number and behavior of the zeros of a random harmonic polynomial asymptotically emulate the same model of random complex analytic polynomial as the degree increases?" We also inspect the variance of the number of zeros of random harmonic polynomials, motivating the work by the question of whether the distribution of the number of zeros concentrates near its as the degree of the harmonic polynomial increases.

Title:	Random Harmonic Polynomials.	126 views 38 downloads
Name(s):	Thomack, Andrew, author Lundberg, Erik, Thesis advisor Florida Atlantic University, Degree grantor Charles E. Schmidt College of Science Department of Mathematical Sciences
Type of Resource:	text
Genre:	Electronic Thesis Or Dissertation
Date Created:	2017
Date Issued:	2017
Publisher:	Florida Atlantic University
Place of Publication:	Boca Raton, Fla.
Physical Form:	application/pdf
Extent:	84 p.
Language(s):	English
Summary:	The study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization. Three prevalent models of signi cant study are the Kostlan model, the Weyl model, and the naive model in which the coe cients of the polynomial are standard Gaussian random variables. A harmonic polynomial is a complex function of the form h(z) = p(z) + q(z) where p and q are complex analytic polynomials. Li and Wei adapted the Kac integral formula for the expected number of zeros to study random harmonic polynomials and take particular interest in their interpretation of the Kostlan model. In this thesis we nd asymptotic results for the number of zeros of random harmonic polynomials under both the Weyl model and the naive model as the degree of the harmonic polynomial increases. We compare the ndings to the Kostlan model as well as to the analytic analogs of each model. We end by establishing results which lead to open questions and conjectures about random harmonic polynomials. We ask and partially answer the question, \When does the number and behavior of the zeros of a random harmonic polynomial asymptotically emulate the same model of random complex analytic polynomial as the degree increases?" We also inspect the variance of the number of zeros of random harmonic polynomials, motivating the work by the question of whether the distribution of the number of zeros concentrates near its as the degree of the harmonic polynomial increases.
Identifier:	FA00004986 (IID), fau:39803 (fedora)
Degree granted:	Dissertation (Ph.D.)--Florida Atlantic University, 2017.
Collection:	FAU Electronic Theses and Dissertations Collection
Note(s):	Includes bibliography.
Subject(s):	Dissertations, Academic -- Florida Atlantic University Random polynomials. Functions. Polynomials.
Held by:	Florida Atlantic University Libraries
Sublocation:	Digital Library
Persistent Link to This Record:	http://purl.flvc.org/fau/fd/FA00004986
Use and Reproduction:	Copyright © is held by the author, with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Use and Reproduction:	http://rightsstatements.org/vocab/InC/1.0/
Host Institution:	FAU
Is Part of Series:	Florida Atlantic University Digital Library Collections.

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