Compute Hilbert function of a monomial ideal

Question

I'd like to know whether there exist easy methods that compute the Hilbert function of a graded $k$-algebra, without computer programs. My homework asks to me to compute the Hilbert function of $R/I$, where $R=k[x_0, \dots, x_5]$ and $$ I = (x_0 x_3, x_0 x_4, x_0 x_5, x_1 x_4, x_1 x_5, x_3 x_5) $$ For me, it is quite difficult to decide what monomials are not in $I$, therefore I thought of compute a free graded resolution of $R/I$, but for me it is hard to calculate syzygies, because there are too many indeterminates. Some suggestions? Thanks to all!

Answer 1

Easy computations, by using the algorithm from the book of Eisenbud, show that the Hilbert series of $R/I$ is $H_{R/I}(t)=\frac{1+3t}{(1-t)^3}$. In order to compute the Hilbert function one uses that $$\frac{1}{(1-t)^3}=\sum_{n\ge 0}{{n+2}\choose{2}}t^n.$$ Of course, any package dedicated to Commutative Algebra can solve your problem immediately.