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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 $A,B,C,D,A\cap B$.

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I don't see how you could do this without also being able to compute at least the generators of $C\cap D$. For instance, if the variables in the generators of $A$ and $B$ are disjoint from the variables in the generators of $C$ and $D$, wouldn't the generators of $I\cap J$ have to include the generators of $C\cap D$? –  joriki Jul 23 '11 at 7:08

3 Answers 3

Look up Gröbner bases. Algorithms based on them are implemented in most symbolic algebra programs. Try for instance Singular.

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I have used Macaulay2 and Macsyma before. This is NOT what I am looking for. I am looking for an abstract expression for the intersection ideal and not for specific examples. –  B M Jul 20 '11 at 16:02

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.)

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According to this comment, in a GCD domain (such as $R$ in your question), $I \cap J = \langle \{ \operatorname{lcm}(a_i,b_j) \} \rangle$, where $\{a_i\}$ are generators of $I$ and $\{b_j\}$ are generators of $J$. And, if $I = A + C$, then the generators of $A$ together with the generators of $C$ generate $I$; similarly for $J$.

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