Primary decomposition of $J = (XZ-Y^2, X^3-YZ)$

In occassions where I need to find the minimal primes associated to an ideal / find a primary decomp., sometimes I can do it just fine, sometimes I find myself completely blind in the search for it.

So Im here asking for general heuristics for this, and also for relevant lemmas and theorems that are frequently used in such situations.

As an example, could someone please explain to me how to find:

• the minimal primes
• a primary decomposition

Of an ideal such as: $$J = ( XZ-Y^2 , X^3-YZ)$$ , to be understood inside the polynomial ring in the variables X Y Z over a field k.

PS: I already looked for some similar questions on this site, but the ones I found were not very clarifying.

• I do not know an answer to your question, but finding a primary decomposition is not easy. For instance, link.springer.com/article/10.1007/BF01231331. In some sense, this is one of the advantages of reducing a problem into a problem of monomial ideals, and primary decompositions for monomial ideals are well-known. – Youngsu Jan 25 at 16:50
• M2 gives {ideal(y^2-x*z,x^2*y-z^2,x^3-y*z), ideal(y,x)} – Jan-Magnus Økland Apr 20 at 7:44
• Thanks. It can help (by the way, how can I compute that on that website?), but I need also to learn how to compute this stuff by bare hands – GLe Apr 20 at 8:12
• R=QQ[x,y,z] I=ideal(x*z-y^2,x^3-y*z) primaryDecomposition I minimalPrimes I – Jan-Magnus Økland Apr 20 at 13:00