Radical computations in Macaulay2

Question

I'm trying to learn how to compute the radical of polynomial ideals in multiple variables over the real numbers in Macaulay2. From what I've gathered in some stack exchange posts([1], [2]), I just need to define a polynomial ring and run the radical command. Now when I do it, I get the following:

Here, I'm trying the basic example of $I = (x^2 + y^2 - 2, y - x^3) \subset \mathbb{R}[x,y]$.

From what I gathered from what I gathered from this answer as well as from the documentation is that there are general algorithms to compute radicals of polynomial ideals - even if they can get slow with increasing numeric complexity in the generators - and that as long as the ground field is characteristic 0, we can do so in Macaulay2. My questions are:

1. Are there any mistakes in my code? If not, am I misunderstanding something about the algorithm / theory around this?

2. More generally, if one has a groebner basis of an ideal, is there a general algorithm for computing the generators of the radical? I know classifying such a basis in general is a difficult question.

EDIT: 3. I should note that I did see this post which claims that one can only generate a subset of the generating set (in general). I'm not quite sure how that connects to the other linked posts which state that there are algorithms which are slow.

Answer 1

If $I$ is an ideal with generators defined over a subring like $\mathbb{Q} \subset \mathbb{R}$ in your case, then in algorithms such as

• Buchberger's algorithm to compute a Groebner basis of $I$, or

• the algorithm to compute generators of the radical $\sqrt{I}$ of a zero-dimensional ideal $I$

all the steps involve only computations over the subring, so for computational purposes you can replace the big ring (your $\mathbb{R}$) by the subring (your $\mathbb{Q}$).

Moreover, these algorithms need to do exact computations in the respective ring, e.g. mainly we need to know for sure what is the leading monomial of a polynomial (not possible if we do not know the coefficients exactly).

So, repeating your calculation with the exact KK = QQ instead of the inexact KK = RR gives the desired result:

i4 : radical I
o4 = ideal(x^2+y^2-2, xy^2-2x+y)
o4 : Ideal of R
i5 : (radical I) == I
o5 = true

For zero-dimensional ideals $I$ like this one there is a simple algorithm to compute generators of the radical $\sqrt{I}$: compute generators $g_1, g_2$ of the intersections $I \cap \mathbb{Q}[x] = \langle g_1 \rangle$ and $I \cap \mathbb{Q}[y] = \langle g_2 \rangle$, let $h_i = g_i / \gcd(g_i, g_i')$ be the square-free part of $g_i$, and then the radical is $\sqrt{I} = I + \langle h_1, h_2 \rangle$.

To compute the intersections $I \cap \mathbb{Q}[x_i]$ one can use elimination (eliminate all variables except for $x_i$, e.g. using an elimination ordering where $x_i$ is the smallest), or use a single Groebner basis $G = [g_1, \ldots, g_r]$ to compute a finite normal basis $$\operatorname{NB}(I) = \mathbb{M} \setminus \operatorname{LM}(I) = \mathbb{M} \setminus \operatorname{LM}(G) = \mathbb{M} \setminus \langle \operatorname{LM}(g_1), \ldots, \operatorname{LM}(g_r) \rangle$$ which is a $\mathbb{Q}$-vector space basis for the quotient $\mathbb{Q}[x,y]/I$, then express powers of $x$ (after multivariate polynomial division by $G$) as vectors with respect to this basis to find a linear dependence $c_0 + c_1 x + c_2 x^2 + \ldots = 0 \mod I$ between those powers, which yields $c_0 + c_1 x + c_2 x^2 + \ldots \in I$ of smallest degree, hence $I \cap \mathbb{Q}[x] = \langle c_0 + c_1 x + c_2 x^2 + \ldots \rangle$, and the same can be done for $I \cap \mathbb{Q}[y]$.