Compute Intersection of Two Ideals

Question

In general, how do we compute the intersection of two monomial ideals? And could someone walk through an example in calculating the intersection of say $(x_1^2x_2, x_2x_4, x_3x_4x_5)\cap(x_1x_3^2, x_2x_4, x_2^3x_5)?$ Thanks!

Answer 1

Those are examples of monomial ideals (this is, they are generated by monomials). There is a lot of literature on how to compute intersections, products, radicals, primary decompositions of these type of ideals.

I'll give you an example of one of the ideas you may use to compute the intersection of two monomial ideals:

Let $I=(xy, x^2)$ and $J = (y^2, x^3)$. Then polynomials in $I$ are the ones that have $0$ coefficient in $1, x$ and $y^j$ for $j \geq 1$.On the other hand, polynomials in $J$ are the ones having coefficient $0$ in $1, x, x^2, y, xy, x^2y$. Therefore, their intersection is the set of polynomials having $0$ coefficient in $1,x,y$, which is generated by $x^2, xy, y^2$.

I know it sounds difficult but now you just have to generalize this idea to $5$ variables... It is tedious but not hard