On the expected number of zeros of a random harmonic polynomial

Abstract

Abstract. We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

On the zeros of random harmonic polynomials: The truncated model

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Motivated by Wilmshurst's conjecture and more recent work of W. Li and A. Wei [17], we determine asymptotics for the number of zeros of random harmonic polynomials sampled from the truncated model, recently proposed by J. Hauenstein, D. Mehta, and the authors [10]. Our results confirm (and sharpen) their (3/2)−powerlaw conjecture [10] that had been formulated on the basis of computer experiments; this outcome is in contrast with that of the model studied in [17]. For the truncated model we also observe a phase-transition in the complex plane for the Kac-Rice density.

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Zeros of harmonic polynomials, critical lemniscates, and caustics

“…This answers a question posed by the first author more than a decade ago. Also this corollary complements the estimates on the expected number of zeros of Gaussian random harmonic polynomials obtained by Li and Wei in [11] and, more recently, by Lerario and Lundberg in [10]. For example, for m = αn, α < 1, the expected number of zeros is ∼ n ( [11]) for the Gaussian harmonic polynomials and ∼ c α n 3/2 [10] for the truncated Gaussian harmonic polynomials.…”

“…Also this corollary complements the estimates on the expected number of zeros of Gaussian random harmonic polynomials obtained by Li and Wei in [11] and, more recently, by Lerario and Lundberg in [10]. For example, for m = αn, α < 1, the expected number of zeros is ∼ n ( [11]) for the Gaussian harmonic polynomials and ∼ c α n 3/2 [10] for the truncated Gaussian harmonic polynomials. Yet, Corollary 1 yields that among all harmonic polynomials the maximal number ∼ αn 2 of zeros occurs with positive probability, thus expanding further the results in [2] for m = 1.…”

“…For a cusp to occur at θ, the condition in Lemma 6 gives that (11) >0 (Ψ(θ − ), Ψ(θ + )) contains lπ for some integer l.…”

On the zeros of random harmonic polynomials: the Weyl model

“…In the context of harmonic polynomials, Li and Wei showed an explicit formula for E[N n,m ] when the coefficients are independent complex Gaussians [9]. Moreover, they showed that if H satisfies (1) H n,m (z) = n j=0 a j z j + m j=0 b j z j with 0 ≤ m ≤ n, where a j and b j are complex Gaussian random variables satisfying:…”