RE: [Macaulay2] intersection of an ideal with a subring

[Douglas Leonard]

Maybe look at all the options on the elimination of variables documentation page.
In your particular example, the following seems to work:

S=QQ[q1,q2,q3,p1,p2,p3,x1,x2,x3,MonomialOrder=>Eliminate 6];
I=ideal(x1-q1^2-q2^2-q3^2,x2-p1^2-p2^2-p3^2,x3-q1*p1-q2*p2-q3*p3);
J=I+ideal(q1*p2-q2*p1,q1*p3-q3*p1,q2*p3-q3*p2)
selectInSubring(1,gens gb J)

________________________________________
From: maca...@googlegroups.com [maca...@googlegroups.com] on behalf of Hans-Christian Herbig [herb...@gmail.com]
Sent: Thursday, August 09, 2012 2:56 PM
To: maca...@googlegroups.com
Subject: [Macaulay2] intersection of an ideal with a subring

Hi,

I wonder whether one can use M2 for the following type of problem
(see below). I couldn't actually find it in the documentation.

I would like to determine the intersection of an ideal with
a subring.

EXAMPLE (related to rotation in QQ^3):

base ring:
R=QQ[q1,q2,q3,p1,p2,p3]

the ideal in R:
I=(q2*p3-q3*p2, q3^*p1-q1*p3, q1*p2-q2*p1)

Now I would like to intersect I with the subring S of R:
QQ[x1,x2,x3]

given by the polynomials
x1=q1*q1+q2*q2+q3*q3
x2=p1*p1+p2*p2+p3*p3
x3=q1*p1+q2?p2+q3*p3

Expected answer:
x3*x3-x1*x2 (Cauchy-Schwarz)

Comment: S is the subring of O(3) invariants in R.

Maybe somebody can give me a hint.

Many thanks and best regards, Hans-Christian Herbig

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[Hans-Christian Herbig]

Hi Doug,

the proposed method turns out to be quite useful.

Thanks a lot, Hans-Christian.

[Hans-Christian Herbig]

I have one question though: Do we have to assume that the polynomials x1,x2,x3

are algebraically independent (as they are in the example)?

[Douglas Leonard]

x1,x2,x3 are being treated as variables in a ring  S, so they need no be independent of each other.

What is happening in your example is that a Gr\"obner basis is being computed and most of it is being ignored, as only the (0,0)th entry of the matrix is of interest to you:

toString gens gb J

matrix {{x1*x2-x3^2, q3*x2-p3*x3, q2*x2-p2*x3, q1*x2-p1*x3, p3*x1-q3*x3,
       p2*x1-q2*x3, p1*x1-q1*x3, q3*p2-q2*p3, p1^2+p2^2+p3^2-x2, q3*p1-q1*p3,
       q2*p1-q1*p2, q1*p1+q2*p2+q3*p3-x3, q1^2+q2^2+q3^2-x1}}

---

From: maca...@googlegroups.com [maca...@googlegroups.com] on behalf of Hans-Christian Herbig [herb...@gmail.com]

Sent: Friday, August 10, 2012 10:53 AM
To: maca...@googlegroups.com
Subject: Re: [Macaulay2] intersection of an ideal with a subring

To view this discussion on the web visit https://groups.google.com/d/msg/macaulay2/-/BidT7rSgjmEJ.