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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  ξ   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 nonconcentration bounds are obeyed.

Local Universality of Zeroes of Random Polynomials