Cyclotomic generating functions

It is a remarkable fact that for many statistics on finite sets ofcombinatorial objects, the roots of the corresponding generating function areeach either a complex root of unity or zero. We call such polynomials\textbf{cyclotomic generating functions} (CGF's). Previous work studied thesupport and asymptotic distribution of the coefficients of several families ofCGF's arising from tableau and forest combinatorics. In this paper, we continuethese explorations by studying general CGF's from algebraic, analytic, andasymptotic perspectives. We review some of the many known examples of CGF's;describe their coefficients, moments, cumulants, and characteristic functions;and give a variety of necessary and sufficient conditions for their existencearising from probability, commutative algebra, and invariant theory. We furthershow that CGF's are ``generically'' asymptotically normal, generalizing aresult of Diaconis. We include several open problems concerning CGF's.