#11457: Witt Vectors
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Reporter: tdupu | Owner: roed
Type: enhancement | Status: new
Priority: major | Milestone: sage-5.11
Component: padics | Resolution:
Keywords: witt vectors, padic, rings | Merged in:
Authors: Taylor Dupuy, David Roe | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Dependencies:
Stopgaps: |
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Comment (by tdupu):
Replying to [comment:14 darij]:
> Hi Taylor,
>
> > 1) Implementing a division by p^n function. This is just a morphism of
modules from p^{n} R to R/ann(p^{n} ) .
> >
> > 2) Implementing a lifting function. If R has the form A/p^n A then we
would like a function which takes elements of A to elements in A/p^{n+1} A
.
>
> I don't think any of these would help compute Witt vector addition over
an arbitrary commutative ring. When you take ghost components over a ring
in which p is a zero-divisor, you lose information; you can't gain it back
by lifting as far as I know.
I need to think if something like this can be well-defined. This isn't
clear to me right now.
>
> What I meant by "big Witt vectors" are Witt vectors that are not
p-typical (Section 9 of Hazewinkel). We should be able to use some of the
extensive symmetric functions implementation we have in Sage (including
Witt coordinates since #14775). While I'd love to play around with Witt-
Burnside, too, I don't know enough about this generalization to implement
it well (though I can learn).
oooh, I didn't know about this. I need to take a look at this. Do you have
any suggestions? I'm not too familiar with what has been done there.
>
> When you say "specifying a particular p", does p mean the polynomial or
the prime? It might be interesting to do it on a generic polynomial,
though I wasn't thinking of that; having addition, negation and
multiplication would already be a wonderful start.
I means p as a prime. I would like to be able to leave p unspecialized (so
you can see formulas with the symbol 'p' rather than an actual prime). I
vaguely recall mathematica being able to do things like this. It is
probably too much to ask.
--
Ticket URL: <http://trac.sagemath.org/ticket/11457#comment:15>
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