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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.

Thank you in advance for your help!

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    $\begingroup$ 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.) $\endgroup$
    – Emil Jeřábek
    Oct 3 at 9:50

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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

M. Sodin, Zeros of Gaussian analytic functions, Mathematical Research Letters 7, 371–381 (2000), Theorem 1.

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.

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  • $\begingroup$ Thank you for your answer and reference.But will not most of the roots lie on the unit circle as n becomes very large? $\endgroup$
    – AgnostMystic
    Oct 3 at 15:44
  • $\begingroup$ @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 $n$. $\endgroup$
    – Alexandre Eremenko
    Oct 3 at 16:10
  • $\begingroup$ And second,I would be highly obliged if you could kindly explain briefly what will happen in the monin polynomials case(assuming othercoeffcients are iid guassian)?Will the leading coefficient in this case affect the distribution of roots or not? $\endgroup$
    – AgnostMystic
    Oct 3 at 16:11
  • $\begingroup$ @AgnosMystic: for your second question I do not have a ready answer. $\endgroup$
    – Alexandre Eremenko
    Oct 3 at 16:14
  • $\begingroup$ @AgnostMystic I think what you're thinking of is that most of the roots lie near the unit circle - but half of them will lie inside and half outside. $\endgroup$
    – Michael Lugo
    Oct 3 at 19:46

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