#6881: Solving conics over polynomial rings.
-------------------------------------+-------------------------------------
Reporter: victor | Owner: lackermans
Type: enhancement | Status: positive_review
Priority: major | Milestone: sage-6.10
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: conic, curve, | Reviewers: Marco Streng
function field | Work issues:
Authors: Lennart Ackermans | Commit:
Report Upstream: N/A | ee31fca60f6045b675dde0186f84f8eb983d8bf6
Branch: | Stopgaps:
public/conics_rational_function_field|
Dependencies: |
-------------------------------------+-------------------------------------
Changes (by mstreng):
* status: needs_review => positive_review
Comment:
All tests pass. Documentation looks good.
The functions do not work perfectly in all cases due to #20003, but after
bypassing {{{squarefree_decomposition}}}, I get:
{{{
sage: K.<t> = PolynomialRing(GF(7))
sage: C = Conic([5*t^2+4, t^2+3*t+3, 6*t^2+3*t+2, 5*t^2+5, 4*t+3,
4*t^2+t+5])
sage: C.has_rational_point()
True
}}}
and
{{{
sage: F = FiniteField(7)
sage: P.<t> = F[]
sage: K = P.fraction_field()
sage: for i in range(50):
c = [K.random_element() for j in range(6)]
C = Conic(c)
C.has_rational_point(point=True)
....:
(False, None)
(False, None)
(False, None)
(False, None)
(True,
((2*t^8 + 5*t^7 + 6*t^6 + 5*t^5 + 4*t^4 + 5*t^2 + 3*t + 2)/(t^8 + 5*t^7 +
5*t^6
+ 4*t^5 + 3*t^4 + 2*t^3 + t^2 + 5*t + 6) : (t^8 + 2*t^7 + t^5 + t^4 +
2*t^3 + 2
*t^2 + 4*t + 4)/(t^8 + 6*t^7 + t^6 + 4*t^5 + t^4 + 2*t^2 + 5*t + 3) : 1))
(False, None)
(False, None)
(False, None)
(True,
((2*t^8 + t^7 + 6*t^6 + 4*t^5 + 6*t^4 + 5*t^2 + 1)/(t^8 + 2*t^7 + 6*t^6 +
3*t^5
+ 3*t^4 + 6*t^3 + 6*t^2 + 6*t) : (2*t^8 + 4*t^7 + 5*t^6 + 3*t^5 + t^4 +
5*t + 1
)/(t^8 + 2*t^7 + 6*t^6 + 3*t^5 + 3*t^4 + 6*t^3 + 6*t^2 + 6*t) : 1))
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(True,
((2*t^4 + 2*t^3 + 3*t^2)/(t^7 + t^6 + 5*t^4 + t^2 + t + 4) : (5*t^7 +
3*t^6 + t
^5 + 5*t^4 + 6*t^3 + 2*t^2 + 4*t)/(t^7 + t^6 + 5*t^4 + t^2 + t + 4) : 1))
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(True,
((t^8 + 5*t^7 + 2*t^6 + 2*t^5 + 3*t^3 + t^2 + 3)/(t^7 + t^6 + 6*t^4 +
5*t^3 + 6
*t^2 + 2*t + 2) : (4*t^5 + 2*t^4 + 5*t^3 + 4*t^2 + 5*t + 2)/(t^4 + 2*t^3 +
5*t^2
+ 4*t + 5) : 1))
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(False, None)
(True,
((2*t^11 + 3*t^10 + 6*t^9 + t^8 + 6*t^7 + 4*t^6 + t^5 + 4*t^4 + 2*t^3 +
3*t + 5
)/(t^11 + 5*t^10 + 5*t^9 + t^8 + t^7 + 6*t^6 + 6*t^5 + 4*t^4 + 6*t^3 +
4*t) : (2
*t^9 + 4*t^8 + 4*t^7 + 6*t^6 + 5*t^5 + 5*t^4 + 4*t^3 + t^2 + 6*t +
1)/(t^10 + 5*
t^8 + 4*t^7 + 2*t^6 + 3*t^5 + 5*t^4 + 6*t^2 + 5*t) : 1))
(False, None)
(False, None)
(False, None)
(False, None)
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/6881#comment:24>
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