#15348: "R.<a> =" syntactic sugar incorrect for EquationOrder and ZZ.extension
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Reporter: emassop | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-7.0
Component: number fields | Resolution:
Keywords: | Merged in:
Authors: Jeroen Demeyer | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jdemeyer/ticket/15348 | e857c9a484132348092e21cd1de0711686b8b7f0
Dependencies: | Stopgaps:
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Comment (by emassop):
My point about wanting `O.<foo> = K.order(-i+1)` to work is moot, because
`.order` doesn't take a names argument, so that the syntactic sugar
doesn't work in this case anyway.
As far as I can tell these patches indeed fix `R.<a> = EquationOrder(f)`
and `R.<a> = ZZ.extension(f)`, where `f` is a polynomial. Also AFAICT,
this is by virtue of `NumberField` using `1, a, a^2, ...` as basis, and
other things normalizing to that basis.
I tried for instance the following to make sure `ring_generators` could
not be distracted:
{{{
from sage.rings.number_field.order import AbsoluteOrder
K.<i> = NumberField(x^2+1)
V, from_v, to_v = K.vector_space()
O = AbsoluteOrder(K, span([to_v(1), to_v(-i)], ZZ))
print O.ring_generators() == [i] # True, so the ring generator is i, not
-i.
}}}
However, these patches do not fix the case of `R<several letters> = ...`.
For instance
{{{
O.<a,b> = EquationOrder([x^2+1, x^2+2])
print O.ring_generators()
}}}
prints `[-b*a - 1, -3*a + 2*b]` which for this patch to work should be
`[a,b]`. (Incidentally, that first and confusing line is used in the
docstring of `EquationOrder`. Can you get rid of that, or comment on its
unexpected consequences for variables `a` and `b`?)
Replying to [comment:17 jdemeyer]:
> Replying to [comment:12 emassop]:
> > I don't really like unconditionally preferring `ring_generators()` in
such a general place as `_first_n_gens()` is. Perhaps this should be
something like `defining_generators()` instead of `ring_generators()`.
This would be set when defining something by generators, and always refer
to the generators from the definition.
>
> Done.
Changing `defining_generators` is indeed dangerous. Perhaps there could be
`defining_generators.set(...)` that would
* populate the cache if it has no value,
* do nothing if the cache and the argument agree,
* populate the cache with an invalid value when the argument and the cache
disagree.
The invalid value would make a call to `defining_generators` raise an
exception. Also, how does the cache behaving with pickling? Does a cache
override get saved?
Finally, does it make sense to define `defining_generators` in
`category_object.pyx` as simply `gens()`, and put the override to
`ring_generators` in `class Order`?
Pain, pain, pain. I think it's okay to pragmatically go with this approach
now.
Some speculation, since my ideas keep changing about this: Perhaps the
preparser should translate `a.<b> = c(d)` to `a,(b,) =
c._call_and_also_return_names(d, names=('b',))`. Then
`_call_and_also_return_names` gets responsible for calling `c` and making
sure the returned generators are the right ones. When `c` is a function,
this can be implemented using a decorator, or next to `c` itself. For
classes `c`, `_call_and_also_return_names` can be a class method of
`CategoryObject` which calls instantiates an object and calls
`_first_n_gens` on it. This would also circumvent the hack of
prepopulating and having to change the cache of `defining_generators`. Of
course this seems like a bigger change than this bug, so let's do the
pragmatic thing first.
--
Ticket URL: <http://trac.sagemath.org/ticket/15348#comment:19>
Sage <http://www.sagemath.org>
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