#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):
Hi Mike,
Replying to [comment:3 mhansen]:
> Stretch is actually something that I needed.
Of course, this is why one implements things. My aim is to have the
Gröbner basis theory of Aschenbrenner and Hillar implemented in Sage, and
actually I have potential applications to 3-dimensional topology in the
background.
And for this, the permutation actions are essential.
But independent from this, I think the implementation should be more
sparse.
> Yep (that's what you can with infinite loops :-) For my application,
doing the arithmetic will the primary bottleneck.
For me, arithmetic will certainly be a bottleneck, too. I mean, it is
about computing Gröbner bases, and if this is done then there still
remains to compute normal forms for really ''really'' huge polynomials.
> I used the existing finite polynomial rings because they already do fast
arithmetic that I (personally) have no desire to duplicate.
Yep. It is very reasonable to try and avoid duplication!
But how? Perhaps it would be better if the f.g. polynomial rings were more
hidden. For example, the elements of an infinite polynomial ring could be
described not simply as a polynomial (say, {{{x100*x101+x90}}}) but as a
pair formed by a polynomial in a smaller ring and an offset, say
{{{(90,x10*x11+x0)}}}, or perhaps by an even smaller polynomial and a
permutation, say {{{(((0,90),(1,100),(2,101)), x1*x2+x0)}}}.
I think a compromise between speed and size makes sense. Or perhaps even
concurrent models, similar to dense vs. sparse matrices.
Since I am rather tired, I don't know if all this makes sense, though...
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5453#comment:4>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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