Experiments on the zeros of harmonic polynomials using certified counting
Complex Variables
2014-06-24 v1 Algebraic Geometry
Probability
Abstract
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.
Cite
@article{arxiv.1406.5523,
title = {Experiments on the zeros of harmonic polynomials using certified counting},
author = {Jonathan D. Hauenstein and Antonio Lerario and Erik Lundberg and Dhagash Mehta},
journal= {arXiv preprint arXiv:1406.5523},
year = {2014}
}