Zeros of harmonic polynomials, critical lemniscates, and caustics

Abstract

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics and the number of zeros of harmonic polynomials. Contents 1. Introduction 1 2. Proofs of Theorems 1, 2 and 3 4 2.1. Proof of Theorem 1 4 2.2. Proof of Theorem 2 6 2.3. Proof of Theorem 3 8 2.4. A remark on Wilmshurst's conjecture 10 3. Geometry of caustics: Proof of Theorem 4 10 4. Construction of non-convex lemniscate 15 References 17

On the zeros of random harmonic polynomials: the Weyl model

“…We have shown that in the case that m = αn+O(1) that the expected number of zeros of a random Weyl polynomial has growth-order 1 3 m 3 2 Moreover, in this paper as well as in Li and Wei's work on the Kostlan model [4], Lerario and Lundberg's work on the truncated model [3] and Thomack's work on the naive model [5] we have the result that when m is fixed the number of zeros grows linearly in n. This raises the question: Is there a class of Gaussian harmonic polynomials such that the expected number of zeros in the m fixed case increases faster than n?…”

“…The lower bound is a consequence of a generalized argument principle. In fact, these bounds are sharp for each n, though for m = 1 the upper bound has been improved to 3n − 2 [6], and it has been conjectured for fixed m that the upper bound is linear in n [5], [8], [15].…”

On the geometry of random lemniscates

“…Remark 4 (Improving the upper bound). As a consequence of Theorem 4 we can prove that the average number of connected components satisfies the upper bound: (28) Eb 0 (Γ) ≤ c 2 n + o(n) with c 2 = 32 − √ 2 56 ≈ 0.5461...…”

“…Critical lemniscates have been studied recently in [28], and in [35] we posed the problem of studying the average number of components of the critical lemniscate associated to a random harmonic polynomial. In the case when p and q are Kostlan polynomials as in [35] of the same degree n, we conjecture the same outcome as in the above corollary-that the average number of components of a critical lemniscate grows linearly with n. Another object of interest is the image under F of the critical set, referred to as the caustic.…”

“…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].…”