In this question, it is discussed that “if the dimension of projective variety $X$ is $n$ and $p_X(m)$ is its Hilbert polynomial, then leading term of $p_X$ is $\dfrac{\deg{X}}{n!}m^n$.”
I might be asking an obvious question, excuse my ignorance, but how do we know that degree of $p_X$ is also $n$?