Probabilistic bounds of random polynomials

Question

This is follow-up question to my previous question about the expected number of roots . I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$, where each coefficient } $a_i$ is drawn independently from a Gaussian distribution. For a given real number } $r > 0$, I am interested in the probability that all roots of $p(z)$ lie within the circle of radius $r$ in the complex plane Any insights or references on this topic would be greatly appreciated Thank you in advance for your help.

Answer 1

I would start with Theorems 4.1 and 4.2 in [1]. A statement of Theorem 4.2 is as follows: if $\nu_n(B(r))$ is the number of zeros within the disk $B(r)$ of radius $r$ when the degree is $n$, then the expected proportion of zeros inside the disk of radius $e^{-s/(2n)}$ is $$ \lim_n E\Big[\frac{\nu_n(B(e^{-s/(2n)}))}{n}\Big] = \frac{(1-e^{-s}(1+s))}{s(1-e^{-s})} = \frac{1}{2} - \frac{s}{12} + o(s). $$ The paper also explains why studying the proportion of roots inside the disk allows by symmetry to study the roots outside the disk, see (1.7)-(1.8).

[1]: Shepp, L.A. and Vanderbei, R.J., 1995. The complex zeros of random polynomials. Transactions of the American Mathematical Society, pp.4365-4384.