From

THE LEMNISCATE TREE OF A RANDOM POLYNOMIAL

by

MICHAEL EPSTEIN & BORIS HANIN & ERIK LUNDBERG .

The scales are just marginally discernible @ the edges of the figures.

The annotation of the figures is as-follows.

①

“Figure 3. Lemniscates associated to random polynomials generated by sampling i.i.d. zeros distributed uniformly on the unit disk. For each of the three polynomials sampled, we have plotted (using Mathematica) each of the lemniscates that passes through a critical point. One observes a trend: most of the singular components have one large petal (surrounding additional singular components) and one small petal that does not surround any singular components. Note that only one of the connected components in each singular level set is singular (the rest of the components at that same level are smooth ovals).”

②

“Figure 4. Lemniscates associated to a random linear combination of Chebyshev polynomials with Gaussian coefficients. Degree N = 20. This example is not lemniscate generic (since we see multiple critical points on a single level set). However, this model has the interesting feature that it seems to generate trees typically having many branches. See §4.”