Let $A$ be a notherian semi-local ring and $m=rad(A)$, the Jacobi radical of $A$. An ideal $I$ is called an ideal of definition of $A$ if $m^n\subseteq I\subseteq m$ for some $n$. Now, for a finitely generated $A$-module $M$, we define the Hilbert polynomial of $M$ with respect to $I$ by:
$X(M,I;n)=l(M/I^n M)$, where $l$ denotes the length of module $M/I^n M$
We know that the degree of this polynomial is independent of choice of ideal of definition, and we denote this degree $d(M)$.
Now, it's said that if there exists an ideal of definiton of $A$ is generated by $r$ elements, then $d(M)\leq r$.
I am not sure why this follows? The number of homogeneous elements of degree $n$ in $r$ elements should be $\lgroup ^{n+r-1}_{r-1} \rgroup$, which is a polynomial of degree $r-1$ in $n$. Hope someone could help. Thanks!