Compute minimal number of generators of subring
Hi all,
Given a polynomial ring $R = k[x_1,...,x_n]$ over a field $k$ (say $\mathbb{Q}$) and a list of polynomials $p_1,..,p_n$ in $R$ (homogeneous say), is Sage capable of computing the minimal number of generators of the subring generated by $p_1,...,p_n$ over $k$?
Next, suppose $I$ is a homogeneous ideal of $R$. Can the same approach be used to compute the minimal number of generators of this subring after being projected to $R / I$ ?
Edit: Sorry for the bad formatting (see source), please help me fix it if you can!
Edit2: For example, let $R = \mathbb{Q} [x,y]$ and let $p_1 = x^2, p_2 = x^2 y, p_3 = x^4 y$. This generates a principal ideal, but as a subring it has 2 generators and it is those 2 generators that I want.
Edit3: In case it is helpful, the case I'm looking at is a ring with an action of a group, and its subring of invariants for which I have generators.
Edit4: Here's a suggested algorithm: sort the generators by degree. Then sequentially add them, and check if they were already in the subring. Does Sage even have suitable subring capabilities?
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Hint: a concrete example would help exploring this question.
For the suggested algorithm you might use the SAGBI functionality in Singular:
If this is sufficient, then I can add an answer showing how to do it from Sage.
Huh I guess so! I don't know Singular syntax - does it work over a base field like QQ? E.g. does sagbiReduce(x4y+(5/2)x6y, A) return 0? And please also let me know how to deal with projecting it to $R/I$ !
Singular works over a field of characteristic zero here, and yes
sagbiReduce(x4y+(5/2)*x6y, A)
returns0
. In SageMath you can do:Unfortunately I don't know how you would deal with $R/I$.