I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are shown below (left: quadratic; right: cubic).
What explains these distributions? In particular: (1) Why the relative paucity of roots
near the origin. (2) Why is the density concentrated in $\frac{1}{2} \le |z| \le 1$?
(3) Why is the cubic distribution sharper than the quadratic?
2026-05-06 05:06:35.1778043995
Distribution of roots of complex polynomials
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