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