The 1950's Mark Kac studied polynomials with independent random normal coefficients in $\mathbb{R}$:
$$ f(z) = \sum_{n = 0}^N a_n z^n $$
He shows we can expect $N_f = \frac{2}{\pi} \log n $ roots in $\mathbb{R}$ arguing
\[ \#\{ a < x < b : f(x) = 0 \} = \frac{1}{2\epsilon} \int_a^b \mathbf{1}\big[-\epsilon < f(x) < \epsilon\big]\,\big|f'(x)\big| \; dx = \mathbb{E}\big[|f'(x)|\; \big| \; f(x) = 0 \big] \]
You can imagine doing this for any distribution on coefficients up to rescaling.
Let $a_i$ be uniform in the unit disk $\mathbb{D} = \{ |z| < 1\}$, $i = 1, \dots, n$.
For arbitrary vectors $\vec{x}, \vec{y}$, you can ask for the conditional expectation:
\[ \mathbb{E} \big[ \vec{a}\cdot \vec{x} \big| \vec{a}\cdot \vec{y} = 0\big] \]
Kac notes he couldn't find a similar paper by Littlewood and Offord from 1938 (didn't have Google!)
Recently this kind of problem has been discussed by Tao (various estimates) or Zelditch (holomorphic sections on Riemann surfaces.
Edelman and Kostlan count solutions for Gaussian random polynomials in a geometric way via the Crofton formula.
We are counting the number of intersections of a curve on the sphere with a random hyperplane.
Finding the zero of a polynomial is like showing two vectors are orthogonal:
$$ \sum a_n z^n = (a_0, a_1, a_2, \dots, a_n) \cdot (1, z, z^2, \dots, z^n) = 0 $$
where $z \in \mathbb{R}$ and $\vec{a} \in \mathbb{R}P^{n+1}=S^n$.
The Gaussian joint distribution of coefficients amounts to the uniform distribution $\vec{a}$ on the sphere.
The Crofton formula says your expected number of real zeros in an interval is equal to the arc length of your curve projected onto the sphere.
$$N_f = \frac{1}{\pi} \int_I \left( \frac{\partial^2}{\partial x \partial y} \log [v(x)\cdot v(y) ]\right)^{\frac{1}{2}}_{x=y=t}dt $$
The Edelman-Kostlan paper has a version of this function for $\mathbb{C}$.
For iid Gaussian coefficients, Peres-Virag work this out to be $\rho(z) = \frac{1}{(1 - |z|^2)^2}$ in the limit that the degree gets large. At least you can see a concentration around the unit circle $|z| = 1$.
For polynomials Edelman-Kostlan propose rescaling - this is related to the $SU(2)$ lie group:
\[ f(z) = \sum_{n=0}^L \binom{L}{k}^{1/2} z^n \]
Then $\mathbb{E}[f(z)\overline{f(w)}] = (1 + z\overline{w})^N$ is covariance for values between two points. This example is discussed by Bleher.
If you are looking for roots of polynomials uniformly scattered in the plane, I recommend recaling the coefficients by $\sqrt{n!}$
$$ f(z) = \sum_{n = 0}^\infty a_n \sqrt{\frac{L^n}{n!}} z^n$$
This is discussed by Krishnapur