#14812: p-adic root finding broken (mathematically incorrect answer)
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Reporter: robharron | Owner: roed
Type: defect | Status: new
Priority: blocker | Milestone: sage-5.12
Component: padics | Resolution:
Keywords: padics, roots | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by roed):
Yikes. This is a precision problem:
{{{
sage: x = polygen(ZZ)
sage: f = x^5 + x^4 - 4*x^2 - x + 3
sage: R = Qp(3,print_mode='digits')
sage: fQp = f.base_extend(R)
sage: h = [a[0] for a in fQp.factor()]
sage: h
[(...1)*x + (...1),
(...1)*x + (...12101021200010200212),
(...1)*x + (...21121201022212022020),
(...1)*x^2 + (...11222222222222222221)*x + (...1000000000000000001)]
sage: h[1]*h[2]
(...1)*x^2 + (...11000000000000000002)*x + (...20000000000000000010)
sage: h[3].discriminant().valuation()
19
sage: prod(h) == fQp
True
}}}
If you cut off the precision at some point then there are multiple valid
factorizations, and this one has the unfortunate feature that it doesn't
find all the roots. `h[3]` is very close to `(x-1)^2`, but its
discriminant is 3^19^, which is not a square.
Note the sensitivity to the precision of the input:
{{{
sage: R19 = Qp(3,19,print_mode='digits',print_pos=False)
sage: fQp19 = f.base_extend(R19)
sage: [a[0] for a in fQp19.factor()]
[(...1)*x + (...1),
(...1)*x + (...121202120222222222),
(...1)*x + (...1001020101222222222),
(...1)*x + (...2201021200010200212),
(...1)*x + (...1121201022212022020)]
}}}
The precision on the two roots near `-1` is a bit disturbing. I'm curious
how things go with #12561 applied, but don't have time to check at the
moment.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14812#comment:1>
Sage <http://www.sagemath.org>
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