def __init__(self, A, f):
r"""
See ``Conic`` for full documentation.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by
x^2 + y^2 + z^2
"""
ProjectiveConic_field.__init__(self, A, f)
def __init__(self, A, f):
r"""
See ``Conic`` for full documentation.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by
x^2 + y^2 + z^2
"""
ProjectiveConic_field.__init__(self, A, f)
def has_rational_point(self,
point=False,
algorithm='default',
read_cache=True):
r"""
Returns True if and only if the conic ``self``
has a point over its base field `F(t)`, which is a field of rational
functions.
If ``point`` is True, then returns a second output, which is
a rational point if one exists.
Points are cached whenever they are found. Cached information
is used if and only if ``read_cache`` is True.
The default algorithm does not (yet) work for all base fields `F`.
In particular, sage is required to have:
* an algorithm for finding the square root of elements in finite
extensions of `F`;
* a factorization and gcd algorithm for `F[t]`;
* an algorithm for solving conics over `F`.
ALGORITHM:
The parameter ``algorithm`` specifies the algorithm
to be used:
* ``'default'`` -- use a native Sage implementation, based on the
algorithm Conic in [HC2006]_.
* ``'magma'`` (requires Magma to be installed) --
delegates the task to the Magma computer algebra
system.
EXAMPLES:
We can find points for function fields over (extensions of) `\QQ`
and finite fields::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18])
sage: C.has_rational_point(point=True)
(True, (-3 : (t + 1)/t : 1))
sage: R.<t> = FiniteField(23)[]
sage: C = Conic([2, t^2+1, t^2+5])
sage: C.has_rational_point()
True
sage: C.has_rational_point(point=True)
(True, (5*t : 8 : 1))
sage: F.<i> = QuadraticField(-1)
sage: R.<t> = F[]
sage: C = Conic([1,i*t,-t^2+4])
sage: C.has_rational_point(point = True)
(True, (-t - 2*i : -2*i : 1))
It works on non-diagonal conics as well::
sage: K.<t> = QQ[]
sage: C = Conic([4, -4, 8, 1, -4, t + 4])
sage: C.has_rational_point(point=True)
(True, (1/2 : 1 : 0))
If no point exists output still depends on the argument ``point``::
sage: K.<t> = QQ[]
sage: C = Conic(K, [t^2, (t-1), -2*(t-1)])
sage: C.has_rational_point()
False
sage: C.has_rational_point(point=True)
(False, None)
Due to limitations in Sage of algorithms we depend on, it is not
yet possible to find points on conics over multivariate function fields
(see the requirements above)::
sage: F.<t1> = FractionField(QQ['t1'])
sage: K.<t2> = FractionField(F['t2'])
sage: a = K(1)
sage: b = 2*t2^2+2*t1*t2-t1^2
sage: c = -3*t2^4-4*t1*t2^3+8*t1^2*t2^2+16*t1^3-t2-48*t1^4
sage: C = Conic([a,b,c])
sage: C.has_rational_point()
Traceback (most recent call last):
...
NotImplementedError: is_square() not implemented for elements of
Univariate Quotient Polynomial Ring in tbar over Fraction Field
of Univariate Polynomial Ring in t1 over Rational Field with
modulus tbar^2 + t1*tbar - 1/2*t1^2
In some cases, the algorithm requires us to be
able to solve conics over `F`. In particular, the following does not
work::
sage: P.<u> = QQ[]
sage: E = P.fraction_field()
sage: Q.<Y> = E[]
sage: F.<v> = E.extension(Y^2 - u^3 - 1)
sage: R.<t> = F[]
sage: K = R.fraction_field()
sage: C = Conic(K, [u, v, 1])
sage: C.has_rational_point()
Traceback (most recent call last):
...
NotImplementedError: has_rational_point not implemented for conics
over base field Univariate Quotient Polynomial Ring in v over
Fraction Field of Univariate Polynomial Ring in u over Rational
Field with modulus v^2 - u^3 - 1
``has_rational_point`` fails for some conics over function fields
over finite fields, due to :trac:`20003`::
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()
Traceback (most recent call last):
...
TypeError: self (=Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + ((6*t^3 + 3*t^2 + 5*t + 5)/(t + 3))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + (t^2 + 3*t + 3)*x*y + (5*t^2 + 5)*y^2 + (6*t^2 + 3*t + 2)*x*z + (4*t + 3)*y*z + (4*t^2 + t + 5)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + ((2*t + 5)/(t + 3))*y + ((3*t^4 + 2*t^3 + 5*t^2 + 5*t + 3)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z : y + ((6*t^3 + 6*t^2 + 3*t + 6)/(t^3 + 4*t^2 + 2*t + 2))*z : z)) domain must equal right (=Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^3 + 6*t^2 + 3*t + 3)*x^2 + (t + 4)*y^2 + (6*t^7 + 2*t^5 + t^4 + 2*t^3 + 3*t^2 + 6*t + 6)*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5/(t^3 + 4*t^2 + 2*t + 2))*x^2 + (1/(t^3 + 3*t^2 + 5*t + 1))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^9 + 5*t^8 + t^7 + 6*t^6 + 3*t^5 + 4*t^3 + t^2 + 5*t + 3))*z^2
Defn: Defined on coordinates by sending (x : y : z) to
((t^3 + 4*t^2 + 2*t + 2)*x : (t^2 + 5)*y : (t^5 + 4*t^4 + t^2 + 3*t + 3)*z)) codomain
TESTS::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: a = (2*t^2 - 3/2*t + 1)/(37/3*t^2 + t - 1/4)
sage: b = (1/2*t^2 + 1/3)/(-73*t^2 - 2*t + 11/4)
sage: c = (6934/3*t^6 + 8798/3*t^5 - 947/18*t^4 + 3949/9*t^3 + 20983/18*t^2 + 28/3*t - 131/3)/(-2701/3*t^4 - 293/3*t^3 + 301/6*t^2 + 13/4*t - 11/16)
sage: C = Conic([a,b,c])
sage: C.has_rational_point(point=True)
(True, (4*t + 4 : 2*t + 2 : 1))
A long time test::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: a = (-1/3*t^6 - 14*t^5 - 1/4*t^4 + 7/2*t^2 - 1/2*t - 1)/(24/5*t^6 - t^5 - 1/4*t^4 + t^3 - 3*t^2 + 8/5*t + 5)
sage: b = (-3*t^3 + 8*t + 1/2)/(-1/3*t^3 + 3/2*t^2 + 1/12*t + 1/2)
sage: c = (1232009/225*t^25 - 1015925057/8100*t^24 + 1035477411553/1458000*t^23 + 7901338091/30375*t^22 - 1421379260447/729000*t^21 + 266121260843/972000*t^20 + 80808723191/486000*t^19 - 516656082523/972000*t^18 + 21521589529/40500*t^17 + 4654758997/21600*t^16 - 20064038625227/9720000*t^15 - 173054270347/324000*t^14 + 536200870559/540000*t^13 - 12710739349/50625*t^12 - 197968226971/135000*t^11 - 134122025657/810000*t^10 + 22685316301/120000*t^9 - 2230847689/21600*t^8 - 70624099679/270000*t^7 - 4298763061/270000*t^6 - 41239/216000*t^5 - 13523/36000*t^4 + 493/36000*t^3 + 83/2400*t^2 + 1/300*t + 1/200)/(-27378/125*t^17 + 504387/500*t^16 - 97911/2000*t^15 + 1023531/4000*t^14 + 1874841/8000*t^13 + 865381/12000*t^12 + 15287/375*t^11 + 6039821/6000*t^10 + 599437/1500*t^9 + 18659/250*t^8 + 1218059/6000*t^7 + 2025127/3000*t^6 + 1222759/6000*t^5 + 38573/200*t^4 + 8323/125*t^3 + 15453/125*t^2 + 17031/500*t + 441/10)
sage: C = Conic([a,b,c])
sage: C.has_rational_point(point = True) # long time (4 seconds)
(True,
((-2/117*t^8 + 304/1053*t^7 + 40/117*t^6 - 1/27*t^5 - 110/351*t^4 - 2/195*t^3 + 11/351*t^2 + 1/117)/(t^4 + 2/39*t^3 + 4/117*t^2 + 2/39*t + 14/39) : -5/3*t^4 + 19*t^3 : 1))
"""
from .constructor import Conic
if read_cache:
if self._rational_point is not None:
return (True, self._rational_point) if point else True
if algorithm != 'default':
return ProjectiveConic_field.has_rational_point(
self, point, algorithm, read_cache)
# Default algorithm
if self.base_ring().characteristic() == 2:
raise NotImplementedError("has_rational_point not implemented \
for function field of characteristic 2.")
new_conic, transformation, inverse = self.diagonalization()
coeff = new_conic.coefficients()
if coeff[0] == 0:
return (True, transformation([1, 0, 0])) if point else True
elif coeff[3] == 0:
return (True, transformation([0, 1, 0])) if point else True
elif coeff[5] == 0:
return (True, transformation([0, 0, 1])) if point else True
# We save the coefficients of the reduced form in coeff
# A zero of the reduced conic can be multiplied by multipliers
# to get a zero of the old conic
(coeff, multipliers) = new_conic._reduce_conic()
new_conic = Conic(coeff)
transformation = transformation \
* new_conic.hom(diagonal_matrix(multipliers))
if coeff[0].degree() % 2 == coeff[1].degree() % 2 and \
coeff[1].degree() % 2 == coeff[2].degree() % 2:
case = 0
def has_rational_point(self, point = False, algorithm = 'default',
read_cache = True):
r"""
Returns True if and only if the conic ``self``
has a point over its base field `F(t)`, which is a field of rational
functions.
If ``point`` is True, then returns a second output, which is
a rational point if one exists.
Points are cached whenever they are found. Cached information
is used if and only if ``read_cache`` is True.
The default algorithm does not (yet) work for all base fields `F`.
In particular, sage is required to have:
* an algorithm for finding the square root of elements in finite
extensions of `F`;
* a factorization and gcd algorithm for `F[t]`;
* an algorithm for solving conics over `F`.
ALGORITHM:
The parameter ``algorithm`` specifies the algorithm
to be used:
* ``'default'`` -- use a native Sage implementation, based on the
algorithm Conic in [HC2006]_.
* ``'magma'`` (requires Magma to be installed) --
delegates the task to the Magma computer algebra
system.
EXAMPLES:
We can find points for function fields over (extensions of) `\QQ`
and finite fields::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18])
sage: C.has_rational_point(point=True)
(True, (-3 : (t + 1)/t : 1))
sage: R.<t> = FiniteField(23)[]
sage: C = Conic([2, t^2+1, t^2+5])
sage: C.has_rational_point()
True
sage: C.has_rational_point(point=True)
(True, (5*t : 8 : 1))
sage: F.<i> = QuadraticField(-1)
sage: R.<t> = F[]
sage: C = Conic([1,i*t,-t^2+4])
sage: C.has_rational_point(point = True)
verbose 0 (3369: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
...
(True, (-t - 2*i : -2*i : 1))
It works on non-diagonal conics as well::
sage: K.<t> = QQ[]
sage: C = Conic([4, -4, 8, 1, -4, t + 4])
sage: C.has_rational_point(point=True)
(True, (1/2 : 1 : 0))
If no point exists output still depends on the argument ``point``::
sage: K.<t> = QQ[]
sage: C = Conic(K, [t^2, (t-1), -2*(t-1)])
sage: C.has_rational_point()
False
sage: C.has_rational_point(point=True)
(False, None)
Due to limitations in Sage of algorithms we depend on, it is not
yet possible to find points on conics over multivariate function fields
(see the requirements above)::
sage: F.<t1> = FractionField(QQ['t1'])
sage: K.<t2> = FractionField(F['t2'])
sage: a = K(1)
sage: b = 2*t2^2+2*t1*t2-t1^2
sage: c = -3*t2^4-4*t1*t2^3+8*t1^2*t2^2+16*t1^3-t2-48*t1^4
sage: C = Conic([a,b,c])
sage: C.has_rational_point()
...
Traceback (most recent call last):
...
NotImplementedError: is_square() not implemented for elements of
Univariate Quotient Polynomial Ring in tbar over Fraction Field
of Univariate Polynomial Ring in t1 over Rational Field with
modulus tbar^2 + t1*tbar - 1/2*t1^2
In some cases, the algorithm requires us to be
able to solve conics over `F`. In particular, the following does not
work::
sage: P.<u> = QQ[]
sage: E = P.fraction_field()
sage: Q.<Y> = E[]
sage: F.<v> = E.extension(Y^2 - u^3 - 1)
sage: R.<t> = F[]
sage: K = R.fraction_field()
sage: C = Conic(K, [u, v, 1])
sage: C.has_rational_point()
...
Traceback (most recent call last):
...
NotImplementedError: has_rational_point not implemented for conics
over base field Univariate Quotient Polynomial Ring in v over
Fraction Field of Univariate Polynomial Ring in u over Rational
Field with modulus v^2 - u^3 - 1
``has_rational_point`` fails for some conics over function fields
over finite fields, due to :trac:`20003`::
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()
...
Traceback (most recent call last):
...
TypeError: self (=Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + ((6*t^3 + 3*t^2 + 5*t + 5)/(t + 3))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + (t^2 + 3*t + 3)*x*y + (5*t^2 + 5)*y^2 + (6*t^2 + 3*t + 2)*x*z + (4*t + 3)*y*z + (4*t^2 + t + 5)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + ((2*t + 5)/(t + 3))*y + ((3*t^4 + 2*t^3 + 5*t^2 + 5*t + 3)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z : y + ((6*t^3 + 6*t^2 + 3*t + 6)/(t^3 + 4*t^2 + 2*t + 2))*z : z)) domain must equal right (=Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^3 + 6*t^2 + 3*t + 3)*x^2 + (t + 4)*y^2 + (6*t^7 + 2*t^5 + t^4 + 2*t^3 + 3*t^2 + 6*t + 6)*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5/(t^3 + 4*t^2 + 2*t + 2))*x^2 + (1/(t^3 + 3*t^2 + 5*t + 1))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^9 + 5*t^8 + t^7 + 6*t^6 + 3*t^5 + 4*t^3 + t^2 + 5*t + 3))*z^2
Defn: Defined on coordinates by sending (x : y : z) to
((t^3 + 4*t^2 + 2*t + 2)*x : (t^2 + 5)*y : (t^5 + 4*t^4 + t^2 + 3*t + 3)*z)) codomain
TESTS::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: a = (2*t^2 - 3/2*t + 1)/(37/3*t^2 + t - 1/4)
sage: b = (1/2*t^2 + 1/3)/(-73*t^2 - 2*t + 11/4)
sage: c = (6934/3*t^6 + 8798/3*t^5 - 947/18*t^4 + 3949/9*t^3 + 20983/18*t^2 + 28/3*t - 131/3)/(-2701/3*t^4 - 293/3*t^3 + 301/6*t^2 + 13/4*t - 11/16)
sage: C = Conic([a,b,c])
sage: C.has_rational_point(point=True)
(True, (4*t + 4 : 2*t + 2 : 1))
A long time test::
sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
sage: a = (-1/3*t^6 - 14*t^5 - 1/4*t^4 + 7/2*t^2 - 1/2*t - 1)/(24/5*t^6 - t^5 - 1/4*t^4 + t^3 - 3*t^2 + 8/5*t + 5)
sage: b = (-3*t^3 + 8*t + 1/2)/(-1/3*t^3 + 3/2*t^2 + 1/12*t + 1/2)
sage: c = (1232009/225*t^25 - 1015925057/8100*t^24 + 1035477411553/1458000*t^23 + 7901338091/30375*t^22 - 1421379260447/729000*t^21 + 266121260843/972000*t^20 + 80808723191/486000*t^19 - 516656082523/972000*t^18 + 21521589529/40500*t^17 + 4654758997/21600*t^16 - 20064038625227/9720000*t^15 - 173054270347/324000*t^14 + 536200870559/540000*t^13 - 12710739349/50625*t^12 - 197968226971/135000*t^11 - 134122025657/810000*t^10 + 22685316301/120000*t^9 - 2230847689/21600*t^8 - 70624099679/270000*t^7 - 4298763061/270000*t^6 - 41239/216000*t^5 - 13523/36000*t^4 + 493/36000*t^3 + 83/2400*t^2 + 1/300*t + 1/200)/(-27378/125*t^17 + 504387/500*t^16 - 97911/2000*t^15 + 1023531/4000*t^14 + 1874841/8000*t^13 + 865381/12000*t^12 + 15287/375*t^11 + 6039821/6000*t^10 + 599437/1500*t^9 + 18659/250*t^8 + 1218059/6000*t^7 + 2025127/3000*t^6 + 1222759/6000*t^5 + 38573/200*t^4 + 8323/125*t^3 + 15453/125*t^2 + 17031/500*t + 441/10)
sage: C = Conic([a,b,c])
sage: C.has_rational_point(point = True) # long time (4 seconds)
(True,
((-2/117*t^8 + 304/1053*t^7 + 40/117*t^6 - 1/27*t^5 - 110/351*t^4 - 2/195*t^3 + 11/351*t^2 + 1/117)/(t^4 + 2/39*t^3 + 4/117*t^2 + 2/39*t + 14/39) : -5/3*t^4 + 19*t^3 : 1))
"""
from constructor import Conic
if read_cache:
if self._rational_point is not None:
return (True, self._rational_point) if point else True
if algorithm != 'default':
return ProjectiveConic_field.has_rational_point(self, point,
algorithm, read_cache)
# Default algorithm
if self.base_ring().characteristic() == 2:
raise NotImplementedError("has_rational_point not implemented \
for function field of characteristic 2.")
new_conic, transformation, inverse = self.diagonalization()
coeff = new_conic.coefficients()
if coeff[0] == 0:
return (True, transformation([1,0,0])) if point else True
elif coeff[3] == 0:
return (True, transformation([0,1,0])) if point else True
elif coeff[5] == 0:
return (True, transformation([0,0,1])) if point else True
# We save the coefficients of the reduced form in coeff
# A zero of the reduced conic can be multiplied by multipliers
# to get a zero of the old conic
(coeff, multipliers) = new_conic._reduce_conic()
new_conic = Conic(coeff)
transformation = transformation \
* new_conic.hom(diagonal_matrix(multipliers))
if coeff[0].degree() % 2 == coeff[1].degree() % 2 and \
coeff[1].degree() % 2 == coeff[2].degree() % 2:
case = 0
else:
case = 1
t, = self.base_ring().base().gens() # t in F[t]
supp = []
roots = [[], [], []]
remove = None
# loop through the coefficients and find a root of f_i (as in
# [HC2006]) modulo each element in the coefficients' support
for i in (0,1,2):
supp.append(list(coeff[i].factor()))
for p in supp[i]:
if p[1] != 1:
raise ValueError("Expected factor of exponent 1.")
# Convert to monic factor
x = p[0]/list(p[0])[-1]
N = p[0].base_ring().extension(x, 'tbar')
R = PolynomialRing(N, 'u')
u, = R.gens()
# If p[0] has degree 1, sage might forget the "defining
# polynomial" of N, so we define our own modulo operation
if p[0].degree() == 1:
mod = t.parent().hom([-x[0]])
else:
mod = N
if i == 0:
x = -mod(coeff[2])/mod(coeff[1])
elif i == 1:
x = -mod(coeff[0])/mod(coeff[2])
else:
x = -mod(coeff[1])/mod(coeff[0])
if x.is_square():
root = N(x.sqrt())
else:
return (False, None) if point else False
# if case == 0 and p[0] has degree 1, we switch to case
# 1 and remove this factor out of the support. In [HC2006]
# this is done later, in FindPoint.
if case == 0 and p[0].degree() == 1:
case = 1
# remove later so the loop iterator stays in place.
remove = (i,p)
else:
roots[i].append(root)
if remove:
supp[remove[0]].remove(remove[1])
supp = [[p[0] for p in supp[i]] for i in (0,1,2)]
if case == 0:
# Find a solution of (5) in [HC2006]
leading_conic = Conic(self.base_ring().base_ring(),
[coeff[0].leading_coefficient(),
coeff[1].leading_coefficient(),
coeff[2].leading_coefficient()])
has_point = leading_conic.has_rational_point(True)
if has_point[0]:
if point:
pt = new_conic.find_point(supp, roots, case,
has_point[1])
else:
pt = True
return (True, transformation(pt)) if point else True
else:
return (False, None) if point else False
# case == 1:
if point:
pt = new_conic.find_point(supp, roots, case)
else:
pt = True
return (True, transformation(pt)) if point else True