Experiments on the zeros of harmonic polynomials using certified   counting
  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. Using this approach, 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. 
Jonathan D. Hauenstein, Antonio Lerario, Erik Lundberg, Dhagash Mehta
math.CV math.AG math.PR