On the zeros of random harmonic polynomials: the truncated model

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.

15 pages, 1 figure. Minor changes in this version include a more detailed proof of Theorem 3

References in corpus (1)

• n-point Gravitational Lenses with 5(n-1) Images

Cited by in corpus (5)

• On the geometry of random lemniscates
• Zeros of a Two-parameter Family of Harmonic Quadrinomials
• The Hermitian Killing form and root counting of complex polynomials with conjugate variables
• Egerváry's theorems for harmonic trinomials
• A note on the expectation of zeros of random harmonic polynomials: The Kac model