Experiments on the Zeros of Harmonic Polynomials Using Certified Counting
Jonathan D. Hauenstein, Antonio Lerario, Erik T. Lundberg, Dhagash B. Mehta · Experimental Mathematics · 2015
Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach, which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha theory to certify the results. We provide new examples of harmonic polynomials having the most extreme number of zeros known so far; we also study the mean and variance of the number of zeros of random harmonic polynomials.