Trac #17895: Computing all roots is faster than computing a single one

The following example comes from comment:27:ticket:16964 via
comment:2:ticket:17886.

{{{
sage: x,y = polygens(QQ,"x,y")
sage: p1 = x^5 + 6*x^4 - 42*x^3 - 142*x^2 + 467*x + 422
sage: p2 = p1(x=(x-1)^2)
sage: p3 = p2(x=x*y).resultant(p2,x).univariate_polynomial()
sage: p4, = [f[0] for f in p3.factor() if f[0].degree() == 80]
sage: ival = CIF((0.77, 0.78), (-0.08, -0.07))
sage: %time z1, = [r for r in p4.roots(QQbar, False) if r in ival]
CPU times: user 1.43 s, sys: 195 ms, total: 1.62 s
Wall time: 1.47 s
sage: %time z2 = QQbar.polynomial_root(p4, ival)
CPU times: user 1min 5s, sys: 212 ms, total: 1min 5s
Wall time: 1min 5s
}}}

The computation for `z1` works reasonably well and completes in under 2
seconds, but the one for `z2` takes over a minute. Which is definitely
wrong, since computing all roots and then choosing the right one should
be ''more'' work, not ''less'' than just computing a single one!

The reason for this is the time spent in the square-free decomposition
(called from `sage/rings/polynomial/complex_roots.py`) which behave
differently depending whether the polynomial is defined with
coefficients in `QQ` or `AA`
{{{
sage: %time _ = p4.squarefree_decomposition()
CPU times: user 807 µs, sys: 0 ns, total: 807 µs
Wall time: 883 µs
sage: %time _ = p4.change_ring(AA).squarefree_decomposition()
CPU times: user 40.1 s, sys: 3.21 ms, total: 40.1 s
Wall time: 40.1 s
}}}
(this somehow independent problem about gcd is tracked at #28199).

With the branch attached the timings become
{{{
sage: sage: %time z1, = [r for r in p4.roots(QQbar, False) if r in ival]
CPU times: user 411 ms, sys: 152 µs, total: 411 ms
Wall time: 410 ms
sage: sage: %time z2 = QQbar.polynomial_root(p4, ival)
CPU times: user 294 ms, sys: 1 µs, total: 294 ms
Wall time: 296 ms
}}}

URL: https://trac.sagemath.org/17895
Reported by: gagern
Ticket author(s): Vincent Delecroix
Reviewer(s): Simon Brandhorst

Author: Release Manager