The closed cone of a rational series is rational polyhedral

For a multivariate power series f  , let Cone(f)Cone(f) denote the cone generated by the exponents of the monomials with nonzero coefficients. Assume that f   is an expansion of a rational function p/qp/q with gcd(p,q)=1gcd(p,q)=1. Then we prove that the closure Cone¯(f) is equal to Cone(p)+Cone(q)Cone(p)+Cone(q). As applications, we show the irrationality of Euler–Chow series of certain algebraic varieties.