Let \(K\) be a field, and let \(R=K[X_1,\ldots,X_n\) be the polynomial ring with \(\deg X_i=d_i\geq 1,\) \(d_i\) an integer. Let \(M\) be a finitely generated graded \(R\)-module. Then \(M=\oplus_k M_k,\) where \(M_k\) is a \(K\)-vector space of finite dimension and \(M_k=0\) for \(k<< 0.\)
Now we consider the Hilbert series of \(M\) \[ H_M(t):=\sum_{r\in\mathbb Z}(\dim_K M_r)t^r. \] The authors give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules.