On the zeros of random harmonic polynomials: the naive model

Andrew Thomack

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arXiv · 1610.02611

On the zeros of random harmonic polynomials: the naive model

Andrew Thomack

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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 $\mathbb{C}$. In this paper we prove that if $m$ is a fixed number, this expectation is asymptotically $n$ as $n\rightarrow\infty$, and if $m=n$ we find a lower and upper bound of order $n\log n$ for this expectation for sufficiently large $n$.

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Andrew Thomack. 2016-10-09. On the zeros of random harmonic polynomials: the naive model. https://arxiv.org/abs/1610.02611

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