  In this paper, we establish some local universality results concerning the
correlation functions of the zeroes of random polynomials with independent
coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n
c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the
$\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to
second order, the coefficients $c_i$ are deterministic, and the degree
parameter $n$ is large. Our results show, under some light conditions on the
coefficients
  $c_i$ and the tails of $\xi_i, \tilde \xi_i$, that the correlation functions
of the zeroes of $f$ and $\tilde f$ are approximately the same. As an
application, we give some answers to the classical question `"How many zeroes
of a random polynomials are real?" for several classes of random polynomial
models.
  Our analysis relies on a general replacement principle, motivated by some
recent work in random matrix theory. This principle enables one to compare the
correlation functions of two random functions $f$ and $\tilde f$ if their log
magnitudes $\log |f|, \log|\tilde f|$ are close in distribution, and if some
non-concentration bounds are obeyed.
