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What is the radical ideal of $(u^2v-a^3,uv^2-b^3,uv-ab)$ in $\mathbb{C}[u,v,a,b]?$

Above all, to learn how to fish, what would be code that I can use to get the radical? I have not worked with macaulay (computational algebra software) before, so what is a good reference to learn about?

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1 Answer 1

up vote 3 down vote accepted

I'm assuming you refer to Macaulay2 (Macaulay is an older version, not much used today).

The following code gives the radical of your ideal:

R = QQ[u,v,a,b]
I = ideal (u^2*v-a^3,u*v^2-b^3,u*v-a*b)
radI = radical I

So, according to Macaulay2, we have $\sqrt{I} = (a^2-ub,va-b^2,uv-ab)$.

Beware, computing radicals can be extremely slow if you have many generators, because the algorithm must compute a Gröbner basis first. However, in this case, the ideal is binomial, and there are extremely efficient algorithms for computing with binomial ideals. (in Macaulay2, the package "BinomialIdeals" does this).

Some (two) references on how to learn Macaulay2:

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Dear @Fredrik, thank you, it is very helpful. –  Ehsan M. Kermani Nov 30 '12 at 16:58
    
By the way, could you add some references for learning how to use Macauley2? –  Ehsan M. Kermani Nov 30 '12 at 22:34
1  
@ehsanmo: Sure. I've added two helpful links. See the post. –  Fredrik Meyer Dec 1 '12 at 10:59

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