On the zeros of random harmonic polynomials: the Weyl model

Abstract

Li and Wei (2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials. 1 arXiv:1710.06906v1 [math.CV]

On the number of roots for harmonic trinomials

“…Recently, the corresponding geometric sectors as in (II) for harmonic trinomials of the form (1.2) with A = 1, B ∈ C \ {0}, C = −1 has been derived in [22]. For references about location, counting, geometry, and lower/uppers bounds for the moduli of roots for harmonic polynomials including probabilistic approaches and numerical experiments, we refer to [1,11,12,13,22,23,24,25,26,27,29,30,31,32,36,37,38,39,40,54,55,56] and the references therein.…”

On the valence of logharmonic polynomials

“…In spite of these counterexamples, it is still seems likely that, in the spirit of Wilmshurst's conjecture, the maximal valence increases linearly in n for each fixed m [22], [19], [30]. The latter result [30] used a nonconstructive probabilistic method, thus connecting the study of the extremal problem with a parallel line of research on the average number of zeros of random harmonic polynomials that had been investigated (for a variety of models) in [26], [25], [39], [40].…”

Zeros of a One-parameter Family of Rational Harmonic Functions

“…Moreover, the conjecture is false in general by constructing some counterexamples (see [5] and [9]). For the number of zeros of harmonic polynomials with more results, we refer to [3,[10][11][12]17,20]. Compared to the result of Khavinson and Swiatek mentioned above, Khavinson and Neumann [6] stated that the rational harmonic function R(z) = p(z) q(z) − z has no greater than 5n − 5 zeros, where p and q are analytic polynomials for n = max(deg p, deg q) ≥ 2.…”