The study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the first signicant progress was made by Kac, finding an integral formula for the expected number of zeros of real zeros of polynomials with real coefficients. This formula as well as adaptations of the formula to complex polynomials and random fields show an interesting dependency of the number and distribution of zeros on the particular method of randomization. Three prevalent models of signicant study are the Kostlan model, the Weyl model, and the naive model in which the coefficients of the polynomial are standard Gaussian random variables.
A harmonic polynomial is a complex function constructed by summing an analytic polynomial with a conjugated analytic polynomial. 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 find 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 findings 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 mean as the degree of the harmonic polynomial increases.
|Commitee:||Kalies, William, Mireles-James, Jason, Schonbek, Tomas|
|School:||Florida Atlantic University|
|School Location:||United States -- Florida|
|Source:||DAI-B 79/08(E), Dissertation Abstracts International|
|Keywords:||Asymptotic, Harmonic polynomial, Kac, Kac-Rice, Random polynomial|
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