Zeros of harmonic polynomials, critical lemniscates and caustics

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.
Comment: 18 pages, 7 figures, Theorem 1 corrected and a remark added