  We investigate the local distribution of roots of random functions of the
form $F_n(z)= \sum_{i=1}^n \xi_i \phi_i(z) $, where $\xi_i$ are independent
random variables and $\phi_i (z) $ are arbitrary analytic functions. Starting
with the fundamental works of Kac and Littlewood-Offord in the 1940s, random
functions of this type have been studied extensively in many fields of
mathematics.
  We develop a robust framework to solve the problem by reducing, via
universality theorems, the calculation of the distribution of the roots and the
interaction between them to the case where $\xi_i$ are gaussian. In this
special case, one can use the Kac-Rice formula and various other tools to
obtain precise answers.
  Our framework has a wide range of applications, which include the most
popular models of random functions, such as random trigonometric polynomials
and all basic classes of random algebraic polynomials (Kac, Weyl, and
elliptic). Each of these ensembles has been studied heavily by deep and diverse
methods. Our method, for the first time, provides a unified treatment of all of
them.
  Among the applications, we derive the first local universality result for
random trigonometric polynomials with arbitrary coefficients. When restricted
to the study of real roots, this result extends several recent results, proved
for less general ensembles. For random algebraic polynomials, we strengthen
several recent results of Tao and the second author, with significantly simpler
proofs. As a corollary, we sharpen a classical result of Erd{\"o}s and Offord
on real roots of Kac polynomials, providing an optimal error estimate. Another
application is a refinement of a recent result of Flasche and Kabluchko on the
roots of random Taylor series.
