Primary decomposition of $I = (x^2, y^2, xy)$

Question

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field.

How to proceed? Are there algorithms to find such decompositions? Where can I find them?

Answer 1

Yes, there are algorithms to find such decompositions in the book "Monomial Ideals" by "Jürgen Herzog-Takayuki Hibi":

(I deleted the proof of uniqueness)

so

$I = (x^2, y^2, xy)= ( y^2, x) \cap (x^2, y)$