#5453: [with patch, needs review] Create a ring for working with polynomials in
countably infinitely many variables
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Reporter: mhansen | Owner: mhansen
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-3.4.1
Component: commutative algebra | Keywords:
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Comment(by SimonKing):
Replying to [comment:8 SimonKing]:
> No, I only have Sage 3.3. Is Sage 3.4 already out? I'll upgrade, or get
the 3.4-sources.
I made it differently: I removed the .rst-part from the patch; after
removing ReST, the rest worked in Sage 3.3...
Anyway. I think the whole story should be made more sparse.
I suggest to use the method {{{change_ring()}}} of polynomial rings in the
background. Then, {{{stretch()}}} could be implemented along these lines:
{{{
sage: R.<x1,y1>=QQ[]
sage: p=R.random_element()
sage: p
-x1^2 - 1/4*y1^2 - 1/2*y1 - 1
sage: S=R.change_ring(names=['x2','y2'])
sage: p=S(p)
sage: p
-x2^2 - 1/4*y2^2 - 1/2*y2 - 1
sage: S=S.change_ring(names=['x4','y4'])
sage: p=S(p)
sage: p
-x4^2 - 1/4*y4^2 - 1/2*y4 - 1
}}}
In that way, one could keep the underlying ring small.
Problem though: How to add things, if the underlying polynomial rings are
different? Probably you know much more about coercion than I do. But I
guess it should be possible to automatically find a common parent for the
summands as follows:
{{{
sage: q=R.random_element()
sage: q
1/3*x1^2 + 1/12*x1*y1 - 3/10*y1^2 + x1
sage: Varnames=set([str(X) for X in p.variables()]+[str(X) for X in
q.variables()])
sage: T=PolynomialRing(QQ,list(Varnames))
sage: T(p)+T(q)
-3/10*y1^2 + 1/12*y1*x1 + 1/3*x1^2 - 1/4*y4^2 - x4^2 + x1 - 1/2*y4 - 1
}}}
This could be done behind the scenes.
In the example above, I did not take care about the monomial order,
though. This needs to be dealt with in a proper implementation.
Cheers,
Simon
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5453#comment:9>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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