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. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection
| Identifer | oai:union.ndltd.org:fau.edu/oai:fau.digital.flvc.org:fau_39803 |
| Contributors | Thomack, Andrew (author), Lundberg, Erik (Thesis advisor), Florida Atlantic University (Degree grantor), Charles E. Schmidt College of Science, Department of Mathematical Sciences |
| Publisher | Florida Atlantic University |
| Source Sets | Florida Atlantic University |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
| Format | 84 p., application/pdf |
| Rights | 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., http://rightsstatements.org/vocab/InC/1.0/ |
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