Statistical Repartition of The Roots
The statistical properties of the roots of a random polynomial have been the subject of several studies. Let
be a random polynomial. If the coefficients ai are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be located on or close to the unit circle.
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula
where
When the coefficients are Gaussian distributed with a non zero mean and variance of σ, a similar but more complex formula is known.
Read more about this topic: Properties Of Polynomial Roots
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