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!

I was first thinking of maybe finding the gcd of both ideals, then finding if there could be any intersections/multiples of the GCDs, since the GCD of an ideal generates the same ideal, but that hasn't really simplified the problem too much. — Brady Tim, Oct 28 '21 at 12:58
Polynomials in multiple variables doe snot form a principal ideal domain, you can't find a generator for those ideals. But the idea is good. Maybe you can try to see is you can find some explicit elements in teh intersection. Can you name some ? — InfiniteLooper, Oct 28 '21 at 13:04
I guess like $x_2x_4$ is clearly in the intersection, since its a generator in both — Brady Tim, Oct 28 '21 at 13:05
Does this answer your question? Question on intersection of ideals. — Elliot Yu, Oct 28 '21 at 13:10

score 0 · Answer 1 · answered Oct 28 '21 at 14:57

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

Compute Intersection of Two Ideals

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