Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|cite|improve this question
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
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
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
Relevant: – Kristoffer Ryhl May 17 '14 at 8:30

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.