In the first part we study critical points of random
polynomials. We choose two deterministic sequences of complex
numbers,whose empirical measures converge to the same probability measure
in complex plane. We make a sequence of polynomials whose zeros are chosen
from either of sequences at random. We show that the limiting empirical
measure of zeros and critical points agree for these polynomials. As a
consequence we show that when we randomly perturb the zeros of a
deterministic sequence of polynomials, the limiting empirical measures of
zeros and critical points agree. This result can be interpreted as an
extension of earlier results where randomness is reduced. Pemantle and
Rivin initiated the study of critical points of random polynomials.
Kabluchko proved the result considering the zeros to be i.i.d. random
variables.
In the second part we deal with the spectrum of products of Ginibre
matrices. Exact eigenvalue density is known for a very few matrix
ensembles. For the known ones they often lead to determinantal point
process. Let X_1,X_2,...,X_k be i.i.d matrices of size nxn whose
entries are independent complex Gaussian random variables. We derive the
eigenvalue density for matrices of the form Y_1.Y_2....Y_n, where each Y_i
= X_i or (X_i)^{-1}. We show that the eigenvalues form a determinantal
point process. The case where k=2, Y_1=X_1,Y_2=X_2^{-1} was derived
earlier by Krishnapur. The case where Y_i =X_i for all i=1,2,...,n, was
derived by Akemann and Burda. These two known cases can be obtained as
special cases of our result.