Comment #9 on issue 2032 by asmeurer: Ability to work with K[x, 1/x] in
Polys
http://code.google.com/p/sympy/issues/detail?id=2032
Well, I don't have any problem with Poly only partially handling 1/x type
generators and raising NotImplementedError when it encounters something it
can't handle. But silently accepting it and then returning what can then
only be interpreted as wrong results is something else, which I think
should be avoided. Unfortunately, due to the nature of domains in Polys, I
fear that it may not be really possible to only have a "half"
implementation (though I could be wrong).
By the way, this would be an awesome thing to include whenever an
implementation of Frac() (issue 2042) comes.
Regarding my algorithm from comment 7, I was close (the spirit was right,
but I had a typo or two). The correct implementation is
def aspoly1t(p, t):
pa, pd = cancel(p.as_basic()).as_numer_denom()
pa, pd = Poly(pa, t), Poly(pd, t)
assert pd.is_monomial
d = pd.degree(t)
one_t_part = pa.slice(0, d + 1) # requires polys11
t_part = pa - one_t_part
t_part = t_part.to_field().quo(pd)
# Compute the negative degree parts. Also requires polys11.
one_t_part = Poly.from_list(reversed(one_t_part.rep.rep),
*one_t_part.gens, domain=one_t_part.domain)
one_t_part = one_t_part.replace(t, z) # z will be 1/t
ans = t_part.as_poly(t, z) + one_t_part.as_poly(t, z)
return ans
In [106]: p1 = random_poly(x, 10, -10, 10)
In [107]: p2 = random_poly(x, 10, -10, 10)
In [108]: p = p1 + p2.subs(x, 1/x)
In [109]: p
Out[109]:
6 8 8 6 6 4 4 9 5 3 3
4 5 6 7 8 9 10
3 - 7⋅x - ─ - ─── + ── - ── + ── - ── - ── - ── + ── + ── - 9⋅x + 10⋅x +
2⋅x + 9⋅x + 4⋅x - 6⋅x - x + 7⋅x
x 10 9 8 7 6 5 4 3 2
x x x x x x x x x
In [110]: aspoly1t(p, x)
Out[110]: Poly(7*x**10 - x**9 - 6*x**8 + 4*x**7 + 9*x**6 + 2*x**5 + 10*x**4
- 9*x**3 - 7*x - 8*z**10 + 8*z**9 - 6*z**8 + 6*z**7 - 4*z**6 - 4*z**5 -
9*z**4 + 5*z**3 + 3*z**2 - 6*z + 3, x, z, domain='ZZ')
In [111]: aspoly1t(p, x).as_basic().subs(z, 1/x) == p
Out[111]: True
Now, some of this is kind of hackish, so if you know of better ways to do
things, feel free to let me know. Otherwise, I am including this in the
relevant part of the Risch Algorithm, as mentioned in comment 0.
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