WOLFRAM|DEMONSTRATIONS PROJECT

Zeros of Random Kac Polynomials

​
set distribution
normal N(0, 1)
cosine of uniform U(0, 2π)
show unit circle
number of coefficients
50
This Demonstration shows that the zeros of random Kac polynomials
P(z)=
n
∑
i=0
a
i
i
z
=
n
∏
i=1
(z-
z
i
)
with independent and identically distributed (i.i.d.) coefficients
a
i
cluster along the complex unit circle as the polynomial degree increases.
Setting the control "set distribution" to "normal
N(0,1)
", the polynomial coefficients
a
i
are distributed according to the standard normal distribution with zero mean and unit standard deviation
a
i
~N(0,1)
[1].
Setting the control "set distribution" to "cosine of uniform
U(0,2π)
", the coefficients
a
i
are the cosines of the values sampled from a uniform distribution
U(0,2π)
.
You can increase the number of coefficients
a
i
computed from the selected probability distribution to get a larger set of complex roots. For more than 100 or so coefficients, you can see that the zeros cluster around the unit circle in the complex plane.