On 29-Aug-08, at 11:47 AM, John Cremona wrote:
>
> If f is a real polynomial then f.roots(RR) gives the real roots, and
> f.roots(CC) gives the complex (including real) roots. Is there a
> fool-proof way of getting at the non-real roots? (You may assume that
> the coefficients of f are exact, perhaps they are rationals, so the
> questions is certainly well-defined.)
>
> But now I need it again in the context of K.embeddings(L) for K a
> number field.
With a little work, Carl Witty's marvelous QQbar implementation could
help. His code will do the "doubling precision" part for you, and you
can ask if any particular root is real -- but I found some bugs doing
this, shown below.
Also, you could probably be certain that roots(QQbar) - roots(AA)
would work.
Nick
{{{
sage: t = (ZZ['x'].0^2 - 2).roots(QQbar)[0][0]
sage: t.real()
[-1.4142135623730952 .. -1.4142135623730949]
sage: t.real() == t
---------------------------------------------------------------------------
AttributeError Traceback (most recent call
last)
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/<ipython
console> in <module>()
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/element.pyx in
sage.structure.element.Element.__richcmp__ (sage/structure/element.c:
4819)()
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/element.pyx in
sage.structure.element.Element._richcmp (sage/structure/element.c:4404)
()
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/element.pyx in
sage.structure.element.Element._richcmp_ (sage/structure/element.c:
4255)()
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/element.pyx in
sage.structure.element.Element._richcmp (sage/structure/element.c:4714)
()
/Users/ncalexan/sage-3.0.6/local/lib/python2.5/site-packages/sage/
rings/qqbar.py in __cmp__(self, other)
2444 """
2445 if self is other: return 0
-> 2446 rcmp = cmp(self.real(), other.real())
2447 if rcmp != 0:
2448 return rcmp
/Users/ncalexan/sage-3.0.6/local/lib/python2.5/site-packages/sage/
rings/qqbar.py in real(self)
2680
2681 def real(self):
-> 2682 return AlgebraicReal(self._descr.real(self))
2683
2684 def imag(self):
/Users/ncalexan/sage-3.0.6/local/lib/python2.5/site-packages/sage/
rings/qqbar.py in real(self, n)
1805
1806 def real(self, n):
-> 1807 if self.is_complex():
1808 return ANUnaryExpr(n, 'real')
1809 else:
AttributeError: 'ANUnaryExpr' object has no attribute 'is_complex'
sage: t.is_complex()
---------------------------------------------------------------------------
AttributeError Traceback (most recent call
last)
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/<ipython
console> in <module>()
AttributeError: 'AlgebraicNumber' object has no attribute 'is_complex'
sage: t
[-1.4142135623730952 .. -1.4142135623730949]
sage: t.real()
[-1.4142135623730952 .. -1.4142135623730949]
sage: t.complex()
---------------------------------------------------------------------------
AttributeError Traceback (most recent call
last)
/Users/ncalexan/Devel/Squeak-3.10-1/platforms/unix/bld/<ipython
console> in <module>()
AttributeError: 'AlgebraicNumber' object has no attribute 'complex'
sage: t.imag()
[0.00000000000000000 .. 0.00000000000000000]
sage: t.imag() == 0
True
}}}
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