elimination_ideal returns an ideal of the original ring

[mezzarobba]

If J is an ideal of a Multivariate Polynomial Ring R, J.elimination_ideal returns an ideal of R, regardless of the variables being eliminated. Is this intentional?

For instance, in the doctest

sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)
sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3]
sage: I.elimination_ideal([t,s])
Ideal (y^2 - x*z, x*y - z, x^2 - y) of Multivariate
Polynomial Ring in x, y, t, s, z over Rational Field

the final result is an ideal of Q[x,y,t,s,z], while I would have expected an ideal of Q[x,y,z].

Component: algebra

Issue created by migration from https://trac.sagemath.org/ticket/10254

[lftabera]

comment:1

I think that this is intended for compatibility with Singular, which is who really computes the elimination ideal. Singular returns an ideal in QQ[x,y,t,s,z], so does SAGE.

• If this is going to change one would need to create the new ring of the elimination ideal and coerce the ideal to this ring.

• One also has to check that both ring have the canonical coercion well defined and that we can operate ideals and polynomials in both rings automatically.

• Also, the code should raise a Warning each time it is called during a transition time.

I am already used to this behaviour and sometimes it is convenient to add further polynomials in the variables of the original ring. So I do not have strong feelings against (or in favour) of this change. It might be more convenient to compute, let's say, dimensions.

[lftabera]

comment:2

I have another idea. We have the intersection method. Used to compute the intersection oftwo ideals. This method could be extended to intersect the ideal with a subring of the form QQ[x1,x2,x3]. This would compute an elimination ideal restricted to the smaller field. Would this approach be convenient?