On the zeros of random harmonic polynomials: the truncated model

TL;DR

This paper analyzes the asymptotic behavior of zeros in a new truncated model of random harmonic polynomials, confirming a conjecture about their power-law distribution through rigorous mathematical results.

Contribution

It provides the first asymptotic analysis for the truncated model, confirming and refining the (3/2)-powerlaw conjecture based on prior computational evidence.

Findings

Asymptotics for the truncated model's zeros were derived.

Confirmed the (3/2)-powerlaw distribution of zeros.

Sharpened previous conjectures with rigorous proofs.

Abstract

A probabilistic approach to the study of the number of zeros of complex harmonic polynomials was initiated by W. Li and A. Wei (2009), who derived a Kac-Rice type formula for the expected number of zeros of random harmonic polynomials with independent Gaussian coefficients. They also provided asymptotics for a complex version of the Kostlan ensemble. Here we determine asymptotics for the alternative truncated model that was recently proposed by J. Hauenstein, D. Mehta, and the authors. Our results confirm (and sharpen) their (3/2)-powerlaw conjecture that had been formulated on the basis of computer experiments.