Dear Sage developers,
consider the following example:
R.<a,b> = GF(5)[]
Q = R.fraction_field()
S.<X,Y> = Q[]
f = 2*R.gen(0)*X*Y^2
w1 = f.coefficient({X:1, Y:2})
w2 = f[1,2]
w3 = f.coefficients()[0]
I would expect w1, w2 and w3 all to be the same thing. However, the
coercions R(w2) and R(w3) work fine, while R(w1) raises an error.
Trying to trace what's wrong, I see that the parent of w1 is S, rather than
Q. While that is what the doc of .coefficient tells, is there any *sane*
reason why the parent of a coefficient isn't the field (or ring) of
coefficients? It seems to me that extracting the coefficient via f[1,2] is
not documented, so I'm a little hesitant to use this as a work around for
the strange behavior of f.coefficient({X:1, Y:2}).
Talking of documentation: the tab completion offers for f and R the method
.base_extend. While I have no idea what that should mean for f (and
f.base_extend? tells nothing), I do have an idea what it could be for R.
Well, the whole entry of R.base_extend? is the truly useless info
EXAMPLES:
sage: QQ.base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no base extension defined
sage: ZZ.base_extend(GF(7))
Finite Field of size 7
without telling what the method is supposed to be. Trying
R.base_extend(GF(25,"dummy")) raises an error, so it doesn't seem to be a
base ring extension. Also, for some time I believed that it is not possible
to change the term order of a polynomial ring, until I learned (from a
recent answer in this forum) that, despite its misleading name,
.change_ring does that. I believe that a major weakness of the sage
documentation is that there are no ``see also''s at the end of a doc entry.
For instance, R.term_order? should point to R.change_ring.
-- Peter Mueller
P.S.: I'm sad to say that while I'm using Sage quite a bit, the lousy
documentation and numerous inconsistencies (like permutation group actions,
with left notation for right action!!!) prevents me and many colleagues
from using Sage in teaching.
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