On the zeros of random harmonic polynomials: the naive model

2016/10/09 by Andrew Thomack, Thomack, Andrew
Mathematics · #30C15 #60G60 #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR) #math.CV #math.PR #msc:30C15 #msc:60G60

paper · pdf · doi:10.48550/arxiv.1610.02611

10 pages

arxiv created 2016/10/09 · arxiv updated 2016/10/11

Abstract

A complex harmonic polynomial is the sum of a complex polynomial and a conjugated complex polynomial, of degrees n and m respectively. Li and Wei (2009) presented a formula for the expected number of zeros of a random harmonic polynomial in ℂ. In this paper we prove that if m is a fixed number, this expectation is asymptotically n as n→∞, and if m=n we find a lower and upper bound of order nlog n for this expectation for sufficiently large n.

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