#16964: Fix comparisons in QQbar
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Reporter: gagern | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.4
Component: algebraic geometry | Resolution:
Keywords: variety qqbar cmp singular | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by gagern):
Looking for ways to address this, I noticed that the a lot of time
apparently is spent inside
{{{
class ANBinaryExpr(ANDescr):
⋮
def exactify(self):
⋮
gen = left._exact_field().union(right._exact_field())
⋮
}}}
I wonder whether we can avoid that union for the case where both fields
have the same defining polynomial. I wonder whether we can assume that the
root of the field will form a
[http://en.wikipedia.org/wiki/Algebraic_number_field#Power_basis power
basis], and if so, whether there is any reasonably cheap way to find the
conversion between different power bases, so we could express one root in
terms of another.
Since I don't have any good ideas how to achieve this, my best idea still
is tacking this at the `__cmp__` level, but if anyone has an idea for
solving this more generic issue, that would be really great since it would
help other computations as well. So I'm sharing my thoughts.
Here is what I've tried and discarded, so you can avoid that same cul de
sac. I started by writing down a generic linear combination w = a₀ + a₁z +
a₂z² + … and then computed p(w) reduced by p(z), where p is the polynomial
of the field. This gives a polynomial in z, and if enough powers of z are
irrational then all the coefficients have to be zero if w is a root and
the a_i are to be rational. So this gave me d conditions on these a₀
through a_{d-1}, which I could combine into an ideal and try to compute a
variety. But that variety computation takes like forever in the above
example, so this generic approach of finding other roots is not feasible
in this fashion. Been there, tried that and discarded it.
--
Ticket URL: <http://trac.sagemath.org/ticket/16964#comment:5>
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