The behaviour of classes of random polynomials

Abstract

This thesis studies the mathematical behavior of random polynomials in terms of the expected number of real zeros and the exceedance measure. To this end, the important formulae for studying this behavior are reviewed, generalized and applied to random polynomials of three different types: the polynomials with symmetric coefficients, the amplified polynomials and the de-amplified polynomials. By defining the coefficients aj’s from the classic form of polyno-mial P (x) = Ln ajxʲ as independently normally distributed random vari-ables with two-fold symmetry, aj = an−j−₁, a polynomial with symmetric coefficients is studied. Then by introducing binomial factors, ⁿ ¹/², into coeffi-cients, the types of amplified and de-amplified polynomials are defined. Besides the expected number of real zeros, the exceedance measure is considered for better understanding of the behavior of amplified and de-amplified polyno-mials. Later in this thesis a valued progress is made towards a conjecture that constants been missed from the sum of binomial series have insignificant impact on the expected number of real zeros. The early studies in this thesis impliesthat the binomial factors facilitate the evaluation of the behavior, therefore amore difficult class of random polynomial without binomial factors is studiedas final part of work of the thesis and supported by numerical analysis.

Thesis is embargoed until 31st October 2014

Date of Award	Oct 2012
Original language	English
Sponsors	Vice Chancellor's Research Scholarship (VCRS)
Supervisor	Kambiz Farahmand (Supervisor) & Mark McCartney (Supervisor)