On Sun, 29 Oct 2006 15:21:30 -0800, David Roe <[EMAIL PROTECTED]> wrote:
>
> Hey all,
> So, I'm making progress on p-adics and power series, though slowly. A
> couple of questions about unifying the terminology.
>
> 1. There are numerous possibilities for talking about the various
> types of precision (eg precision, relative precision, modulus,
> big_oh...). There are two concepts that we need to label with these
> terms: the precision of the unit part and the ideal of the ring that
> we're working modulo. For example, 3^3+3^5+O(3^7) would have
> precisions 4 and 7. I have been using the terms "precision" for the
> first part and "modulus" for the second, but I'm not convinced that
> this is the right choice because the term "precision" is fairly well
> entrenched in the power series world as the second. So if "precision"
> is the second, perhaps we can use "relative precision" for the first.
> What do people think?
I like using relative_precision and absolute_precision (following what
I'm used to from MAGMA). Then there is no confusion.
> 2. A second issue arises with extension fields. In a tower of fields
> (or rings), we want to have functions that return the field immediately
> below the current one, as well as the field at the bottom of the tower.
> We were considering using "ground field" and "base field" ("ground
> ring" and "base ring"), but base ring is used in power series to mean
> the coefficient ring. How about "ground field" and "rock bottom field"
> ("ground ring" and "rock bottom ring")? Also, should the function
> names in the field and ring cases differ or should we call them both
> ring extensions?
I don't like "rock bottom field" or "rock bottom ring", since they are
not terms used in mathematics. For base ring I always define base_ring,
and I define base_field when it happens to also be a field. But they
should be synonyms if they are both defined.
Base ring/field should always be the previous step in the tower.
Maybe the function for the base first object in a tower of extensions
could be called
base_ring_of_tower()
and/or
base_field_of_tower()
This has the advantage that you would find it very easily via tab
completion, and it sounds mathematical and clean.
> 3. There are a couple of functions in the current implementations of
> p-adics and power series that I don't think make all that much sense to
> be there, given their nonexact nature (also because I'm making power
> series immutable). In particular, I would like to delete
> additive_order, multiplicative_order, is_zero from padics and
> __setitem__ from power series.
Definitely delete __setitem__ -- I am making all ring elements immutable,
so setitem should go (I've probably already deleted it in my copy).
additive_order, multiplicative_order, and is_zero all make perfect
sense for p-adics, so i don't want to delete them. You could require
that each function takes a non-optional precision argument, and claim
to only compute the answer using information up to that precision.
E.g., is_zero would mean zero up to that precision. additive_order
would be oo unless is_zero(prec) is True. multiplicative_order
would involve working modulo p^n, maybe.
William
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