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 $S/ \langle f_1,\cdots,f_q\rangle$?
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 $S/ \langle f_1,\cdots,f_q\rangle$?
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?