On the zeros of random harmonic polynomials: The truncated model

Abstract

Abstract. 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.

Asymptotic variance of the number of real roots of random polynomial systems

“…, and zero for odd indices [10]. Introducing now the distribution m j=1 δ 0 (y j ) and denoting as b α its coefficients it holds 11) or b α = 0 if at least one index α j is odd.…”

“…His results include the finiteness of the limit variance, when the degree tends to infinity, of the volume of the zero sets of Kostlan-Shub-Smale systems with strictly less equations than variables. Some results for the expectation and variance of related models are included in [2,11,12].…”

On the zeros of random harmonic polynomials: the Weyl model

“…As written, this theorem is more general than as it is written in [9] and [7], by replacing specific variances of a j and b j with arbitrary positive values α j and β j , though the same proof suffices.…”

“…where c α is a constant depending only on α ∈ (0, 1] [7]. Moreover, c α → π 4 as α → 1 giving the asymptotic value of E[N n,m ] a satisfying continuity in α, a property shared by the particular model of random H that is the focus of this article.…”

On the geometry of random lemniscates

“…Lemniscates appear in a variety of specific studies and applications including approximation theory (e.g. Hilbert's lemniscate theorem and its generalizations [49,39]), topology of real algebraic curves [3,9,27], elliptic integrals from classical mechanics [2], holomorphic dynamics [38, p. 151], numerical analysis [46], operator theory [47], so-called "fingerprints" of twodimensional shapes [10,51], moving boundary problems [29,36], as critical sets of planar harmonic mappings [28,35], and in the theory and applications of conformal mapping [4,22,26].…”