A few days ago I wrote about the expected number of roots in a random polynomial where each coefficient is drawn from a standard normal, i.e. a Gaussian distribution with mean 0 and variance 1.

Another class of random polynomials, one that comes up in applications to physics, draws each coefficient from a different distribution. Specifically, the *n*th coefficient in an *N*th degree polynomial is drawn from a normal distribution with mean zero and variance *N* choose *n*. As before, our reference is [1].

It turns out that for these random polynomials, the expected number of real zeros is √*N*. This is not an asymptotic approximation.

For an example, I created a random 44th degree polynomial.

from scipy.stats import norm from scipy.special import binom N = 44 coeff = [norm.rvs(scale=binom(N,n)**0.5) for n in range(N+1)]

I plotted the (tanh of) this polynomial as described here.

This polynomial clearly has 8 real roots. The expected number of roots for a 44th degree random polynomial of this type is √44 = 6.63.

[1] Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real? Bulletin of the AMS. Volume 32, Number 1, January 1995