#8987: Add support for rational polyhedral fans
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Reporter: novoselt | Owner: mhampton
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.4.4
Component: geometry | Keywords:
Author: Andrey Novoseltsev | Upstream: N/A
Reviewer: Volker Braun | Merged:
Work_issues: |
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Comment(by novoselt):
Replying to [comment:7 vbraun]:
> Maybe I'm missing something, but is there a reason why comparison of
cones depends on the ray ordering?
{{{
sage: Cone([(1,0), (0,1)]) == Cone([(1,0), (0,1)])
True
sage: Cone([(1,0), (0,1)]) == Cone([(0,1), (1,0)])
False
}}}
> It seems more natural to compare `ray_set()` instead of `rays()`.
This is done like this:
{{{
sage: Cone([(1,0), (0,1)]).is_equivalent(Cone([(0,1), (1,0)]))
True
}}}
One reason for such behaviour of `==` is simplicity of comparison, which
probably should be fast for sorting purposes (and I was implementing only
`__cmp__` for all classes, not `__eq__`). In the case of not strictly
convex cones equality of ray sets is not necessary for equality of cones
as sets of points (see the code of `is_equivalent` for cones).
Another reason is consistency with the implementation of fans. For fans
checking mathematical equality is a bit more tedious (although still can
be done quickly using sorted lists for rays and cones, see `is_equivalent`
for fans), but the main reason is that I would prefer to be able to use
plain integers to index divisors or charts. I was participating in the
discussion how the equality of fans should be interpreted in Macaulay2,
and in the documentation to one of the cohomology related functions there
was a line about "canonical identification with `ZZ^n`". Well, such
identifications can be canonical only if the order of things is fixed.
Maybe these arguments are not very strong, but since there is a way to
check equality in both senses, I think that it should be fine. The plan is
also to have equivalence check relevant for toric varieties, i.e. up to
`GL(ZZ^n)` action, but `is_isomorphic` functions throw
`NotImplementedError` so far.
Would you like me to add some comments on this behaviour to the "main"
documentation of cone and fan modules? It is already described in
`is_equivalent` and `is_isomorphic` docstrings, but perhaps deserves more
visibility.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8987#comment:8>
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