radical ideal using macaulay software?

Question

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?

Answer 1

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:

• The Macaulay2 homepage. Here are four guides that will teach you the basics. Follow them step-by-step.
• The Macaulay2 book Computations in Algebraic Geometry with Macaulay2. Lots of examples. And the whole book is available free in all sorts of formats.