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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).
Poly23
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?

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It's probably significant that the distribution of the product of the roots, and therefore that of the product of their absolute values, does not depend on the degree. –  Niels Diepeveen Aug 17 '13 at 16:31
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Have you tried plotting the logarithm of the roots? This seems to show a density that is independent of the imaginary part and symmetric w.r.t. the real part. –  Niels Diepeveen Aug 18 '13 at 2:39
    
@NielsDiepeveen: Nice insight! But I still don't understand what's driving these distributions... –  Joseph O'Rourke Aug 18 '13 at 14:01
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The distribution of roots is invariant under rotation. More precisely, under the transform $a_k \to e^{ik\theta} a_k$, a root $r$ of the polynomial $z^n + a_1 z^{n-1} + \cdots + a_n$ corresponds to a root $r e^{-i\theta}$ of the transformed polynomial. That's why the density is independent of the imaginary part of log of the root. –  achille hui Aug 19 '13 at 12:19
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Relevant: math.ucr.edu/home/baez/week285.html –  Darksonn May 17 at 8:30
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