TITLE: Computing ideal intersections in polynomial rings
QUESTION [7 upvotes]: Suppose, $R=k[x_1,...,x_n]$ and $I,J,A,B,C,D$ are ideals in $R$. Suppose, I can write $A,B,C,D$ explicitly in terms of generators and I can also compute $A\cap B$ explicitly in terms of generators. It is also known,
$I=A+C$
$J=B+D$
How would I go about computing $I\cap J$, if this can be done at all.
To clarify what I mean by explicitly, I can write an ideal as $(f_1,...,f_n)$, where the polynomials $f_i$ are not specified, but I know certain properties of these (so I can write down examples). So, I would like to write down $I\cap J$ in terms of the generators of $A,B,C,D,A\cap B$.

REPLY [2 votes]: I don't know if it's what you are looking for, but for any ideals $I,J\in k[x_1,\ldots,x_n]$ you may write
\begin{align*}
 I\cap J = (tI + (1-t)J) \cap k[x_1,\ldots,x_n],
\end{align*}
where $t\in K[t]$ is a variable, and the notation $tS$ for any ideal $S$ means $\{tf:f\in S\}$.
If you have access to Ideals, Varieties and Algorithms by Cox, Little and O'Shea, it's Theorem 11 in section 4.3 (page 187-188 if you get it on google books.)