Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
You can compute the Hilbert series (and consequently, the Hilbert function) by using an algorithm described by Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Section 15.1.1. – user26857 Jul 18 '12 at 20:30
Thank you very much! – Andrea Sep 15 '12 at 19:19
up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.