Sendov illustration

Motivated by this blog entry: Some pictures: A random polynomial of degree 20 with 20 roots in the unit disc. The picture also illustrates Poincaré-Hopf nicely as the roots have index 1 and the critical points have index -1 (saddles) and because there are 20 roots and 19 critical points by the fundamental theorem of algebra. Already when looking at pictures of degree 80, we observed numerical problems in finding the roots of the derivative: example even when adding a few hundred more extra precision digits. The following picture is for degree 60 and 1000 digit extra precision (standard calculators use 16 digits) We see that critical points behave like ``moons" around planets in the sense that they tend to pair up with roots. We also see that the critical points tend to be more inside and in particular are inside the unit disc. It would be nice to know the radial distribution of the critical points if the distribution of the roots is uniform and random (of course, the critical point distribution is then also rotationally symmetric). The ``moon-planet" picture makes one suspect that the distribution of the critical points in the n to infinity limit is always equal to the prescribed distribution of the roots.
Oliver Knill, Posted December 9, 2020,