It has been shown that zeros of Kac polynomials
of degree
cluster asymptotically
near the unit circle as
under some assumptions. This property remains unchanged for the
-th derivative of the
Kac polynomials
for any fixed order
.
So it’s natural to study the situation when the number of the derivatives we take depends
on
,
i.e.,
.
We will show that the limiting behavior of zeros of
depends on the
limit of the ratio
.
In particular, we prove that when the limit of the ratio is strictly positive, the
property of the uniform clustering around the unit circle fails; when the ratio is close
to 1, the zeros have some rescaling phenomenon. Then we study such problem for
random polynomials with more general coefficients. But things, especially the
rescaling phenomenon, become very complicated for the general case when
,
where we compute the case of the random elliptic polynomials to illustrate
this.
Keywords
derivatives of random polynomials, empirical measure
