Experiments on the zeros of harmonic polynomials using certified counting
arXiv:1406.5523
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.
References in corpus (5)
- Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method
- Phase transitions detached from stationary points of the energy landscape
- Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces
- Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
- Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model