Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all finitely generated $R_0$-modules. Then by Hilbert–Serre the Poincaré series of $M$ is $$P(M,t) = \sum_{n = 0}^{\infty} \lambda(M_n)t^n = \frac{f(t)}{\prod_{i = 1}^{s}(1-t^{k_i})} \in \mathbb{Q}(t)$$ where $x_i \in R$ are homogeneous generators as $R_0$-algebra and $k_i = \deg x_i$ for $i = 1,\dotsc,s$ and $f \in \mathbb{Z}[t]$. $d(M)$ is defined to be the order of the pole at $1$ of $P(M,t)$.
Why is $d(M)\geq 0$? Why does $f(t)$ not have a zero of order $> s = \operatorname{ord}(\prod_{i = 1}^{s}(1-t^{k_i}), 1)$ in $1$? Reference is Atiyah–MacDonald Chapter 11.
