Experiments on the Zeros of Harmonic Polynomials Using Certified Counting

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. 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.

Keywords:

• harmonic polynomials
• Wilmshurst’s conjecture
• homotopy continuation
• alpha-certification
• root-finding

2000 AMS Subject Classification::

• 14Q99
• 65G20
• 65H04

Notes

¹We choose a = 1 + 0.04i/n in the simulations because it appeared to produce a few more zeros than a constant choice, at least when ℓ is small compared to n.

²When m = αn, the numbers become much larger than .