# Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients

I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial

$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^n$$

where each coefficient $$a_i$$ is a standard Gaussian random variable, I am interested in understanding how many of the roots of this polynomial, on average, will lie inside the unit disc in the complex plane, i.e., have a magnitude less than 1.

Formally, let $$C_n$$ denote the set of roots of the polynomial and $$|C_n|$$ denote the number of such roots. I want to find the expected value of the fraction of roots in the unit disc:

$$\mathbb{E} \left[ \frac{| \{ z \in C_n : |z| < 1 \} |}{|C_n|} \right]$$

Any insights, references, or solutions regarding this problem will be greatly appreciated.

• Since the distribution is symmetric under the reversal $(a_0,\dots,a_n)\mapsto(a_n,\dots,a_0)$, which corresponds to taking inverse of each root, the probability has to be $1/2$. (I’m assuming here that with probability $1$, $a_0\ne0\ne a_n$, $f$ has no roots on the unit circle, and $f$ has no repeated roots. These should be easy to establish.) Oct 3 at 9:50

You did not fully specify the distribution of coefficients. Usually in such problems coefficients are assumed independent. Under this assumption, there is a general formula due to Edelman and Kostlan: $$E(D)=\frac{1}{2\pi}\int_D\Delta\log\|\Psi(z)\|dxdy,$$ where $$E(D)$$ is the expected number in zeros in $$D$$ and $$\|\Psi(z)\|^2=1+|z|^2+\ldots+|z|^{2n}=\frac{1-|z|^{2n+1}}{1-|z|^2}.$$ A convenient reference is
But in your case, you do not need this general formula, because it is clear from symmetry that $$1/2$$ of all zeros is expected to be in the unit disk, since the thansformation $$P(z)\mapsto z^nP(1/z)$$ sends your family into itself and preserves your probability measure.
• @AgnosMystic: No. This follows from Edelman and Kostlan formula: it shows that the distribution of expectation is continuous in the plane. This is for fixed $n$, of course. To see what happens for $n\to\infty$, one has to compute the integral in the Edelman-Kostlan formula (it is elementary) and see how the distributuion changes with $n$. Oct 3 at 16:10