TITLE: Distribution of roots of complex polynomials
QUESTION [32 upvotes]: 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?

REPLY [3 votes]: The first two observations can be explained by the fact that if $r$ is the root of $a_nz^n+\cdots +a_0$ then $r^{-1}$ is the root of $a_0z^n+\cdots +a_n$.  Since the joint probability density of the coefficients is symmetric under $a_k\mapsto a_{n-k}$ transformation the density of the roots should be symmetric under $r\mapsto r^{-1}$.  It should be fairly straightforward to find asymptotic behavior of the density as $z\to0$ or $z\to\infty$. In the first case, $r\to a_0/a_1$ and the second case is symmetric under inversion.
The distribution around $|z|=1$ is less obvious, so I need to think more to explain your third observation.