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Let $S=k[X_1,\cdots,X_n]$ and $\{f_1,\cdots f_q\}$ be a $S$-regular sequence with ${\rm deg}(f_i)=a_i$.

What is Hilbert polynomial of $S/ \langle f_1,\cdots,f_q\rangle$?

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    Hint: The Koszul complex of a regular sequence is exact.2012-09-04
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    Thanks David, but I know it, I can't easily formulate it's Hilbert polynomial.2012-09-04
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    Hint 2: Can you compute the Hilbert polynomials of the other terms in the Koszul complex?2012-09-04
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    sorry, david, I don't understand your hint. maybe induction on q?2012-09-05
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    That can be made to work as well. Can you do $q=1$? Can you do $q=2$? If you can do $q=2$, I think you should be able to see the rest.2012-09-05
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    I See $q=1$ case. but In case $q=2$, I can't calculate combinations.2012-09-06

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Let's look at the case $q=2$. To be concrete, I'll take $n=4$. So we have two polynomials $f$ and $g$, in variables $w$, $x$, $y$, $z$. Let $\deg f = a$ and $\deg g=b$. The ring $S$ itself has $\binom{d+3}{3}$ monomials in degree $d$, so the Hilbert polynomial of $S$ is $\binom{d+3}{3}$. I'll write $S_d$ for the degree $d$ part of $S$.

We have the short exact sequence $$0 \to S \stackrel{\cdot f}{\longrightarrow} S \to S/f \to 0$$ and hence $$0 \to S_{d-a} \to S_d \to (S/f)_d \to 0.$$ So $\dim (S/f)_d = \binom{d+3}{3} - \binom{d-a+3}{3}$.

Now, we also have the short exact sequence $$0 \to S/f \stackrel{\cdot g}{\longrightarrow} S/f \to S/\langle f,g \rangle \to 0.$$ (This is where we use that $(f,g)$ is a regular sequence.) Can you finish it from here?

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    Thanks, Daivd. I can do until above step. continuing above step, then $q+1$ combination appear. maybe it's combination can be expressed more simply?2012-09-07
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    So, just to confirm, for any $q$ and any degrees $\deq f_1$, ..., $\deg f_q$, you could find the Hilbert polynomial, and your question is simply whether there is a simpler way to write your answer? The short answer here is "generating functions".2012-09-07
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    Sorry, David, I don't know generating function very well. un...how do I apply the generating function?2012-09-08