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r""" 

Projective plane conics over a number field 

 

AUTHORS: 

 

- Marco Streng (2010-07-20) 

 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2009/2010 Marco Streng <marco.streng@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from sage.rings.all import (RDF, CDF, AA, RLF, QQbar, PolynomialRing) 

 

from sage.rings.complex_field import is_ComplexField 

 

from sage.rings.ring import is_Ring 

from sage.rings.rational_field import is_RationalField 

from sage.rings.real_mpfi import is_RealIntervalField 

from sage.rings.complex_interval_field import is_ComplexIntervalField 

 

from .con_field import ProjectiveConic_field 

 

class ProjectiveConic_number_field(ProjectiveConic_field): 

r""" 

Create a projective plane conic curve over a number field. 

See ``Conic`` for full documentation. 

 

EXAMPLES:: 

 

sage: K.<a> = NumberField(x^3 - 2, 'a') 

sage: P.<X, Y, Z> = K[] 

sage: Conic(X^2 + Y^2 - a*Z^2) 

Projective Conic Curve over Number Field in a with defining polynomial x^3 - 2 defined by X^2 + Y^2 + (-a)*Z^2 

 

TESTS:: 

 

sage: K.<a> = NumberField(x^3 - 3, 'a') 

sage: Conic([a, 1, -1])._test_pickling() 

""" 

def __init__(self, A, f): 

r""" 

See ``Conic`` for full documentation. 

 

EXAMPLES :: 

 

sage: Conic([1, 1, 1]) 

Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2 

""" 

ProjectiveConic_field.__init__(self, A, f) 

 

# a single prime such that self has no point over the completion 

self._local_obstruction = None 

# all finite primes such that self has no point over the completion 

self._finite_obstructions = None 

# all infinite primes such that self has no point over the completion 

self._infinite_obstructions = None 

 

 

def has_rational_point(self, point = False, obstruction = False, 

algorithm = 'default', read_cache = True): 

r""" 

Returns ``True`` if and only if ``self`` has a point 

defined over its base field `B`. 

 

If ``point`` and ``obstruction`` are both False (default), 

then the output is a boolean ``out`` saying whether ``self`` 

has a rational point. 

 

If ``point`` or ``obstruction`` is True, then the output is 

a pair ``(out, S)``, where ``out`` is as above and: 

 

- if ``point`` is True and ``self`` has a rational point, 

then ``S`` is a rational point, 

 

- if ``obstruction`` is True, ``self`` has no rational point, 

then ``S`` is a prime or infinite place of `B` such that no 

rational point exists over the completion at ``S``. 

 

Points and obstructions are cached whenever they are found. 

Cached information is used for the output if available, but only 

if ``read_cache`` is True. 

 

ALGORITHM: 

 

The parameter ``algorithm`` 

specifies the algorithm to be used: 

 

- ``'rnfisnorm'`` -- Use PARI's rnfisnorm 

(cannot be combined with ``obstruction = True``) 

 

- ``'local'`` -- Check if a local solution exists for all primes 

and infinite places of `B` and apply the Hasse principle. 

(Cannot be combined with ``point = True``.) 

 

- ``'default'`` -- Use algorithm ``'rnfisnorm'`` first. 

Then, if no point exists and obstructions are requested, use 

algorithm ``'local'`` to find an obstruction. 

 

- ``'magma'`` (requires Magma to be installed) -- 

delegates the task to the Magma computer algebra 

system. 

 

 

EXAMPLES: 

 

An example over `\QQ` :: 

 

sage: C = Conic(QQ, [1, 113922743, -310146482690273725409]) 

sage: C.has_rational_point(point = True) 

(True, (-76842858034579/5424 : -5316144401/5424 : 1)) 

sage: C.has_rational_point(algorithm = 'local', read_cache = False) 

True 

 

Examples over number fields:: 

 

sage: K.<i> = QuadraticField(-1) 

sage: C = Conic(K, [1, 3, -5]) 

sage: C.has_rational_point(point = True, obstruction = True) 

(False, Fractional ideal (-i - 2)) 

sage: C.has_rational_point(algorithm = "rnfisnorm") 

False 

sage: C.has_rational_point(algorithm = "rnfisnorm", obstruction = True, read_cache=False) 

Traceback (most recent call last): 

... 

ValueError: Algorithm rnfisnorm cannot be combined with obstruction = True in has_rational_point 

 

sage: P.<x> = QQ[] 

sage: L.<b> = NumberField(x^3-5) 

sage: C = Conic(L, [1, 2, -3]) 

sage: C.has_rational_point(point = True, algorithm = 'rnfisnorm') 

(True, (5/3 : -1/3 : 1)) 

 

sage: K.<a> = NumberField(x^4+2) 

sage: Conic(QQ, [4,5,6]).has_rational_point() 

False 

sage: Conic(K, [4,5,6]).has_rational_point() 

True 

sage: Conic(K, [4,5,6]).has_rational_point(algorithm='magma', read_cache=False) # optional - magma 

True 

 

TESTS: 

 

Create a bunch of conics over number fields and check whether 

``has_rational_point`` runs without errors for algorithms 

``'rnfisnorm'`` and ``'local'``. Check if all points returned are 

valid. If Magma is available, then also check if the output agrees with 

Magma. :: 

 

sage: P.<X> = QQ[] 

sage: Q = P.fraction_field() 

sage: c = [1, X/2, 1/X] 

sage: l = Sequence(cartesian_product_iterator([c for i in range(3)])) 

sage: l = l + [[X, 1, 1, 1, 1, 1]] + [[X, 1/5, 1, 1, 2, 1]] 

sage: K.<a> = QuadraticField(-23) 

sage: L.<b> = QuadraticField(19) 

sage: M.<c> = NumberField(X^3+3*X+1) 

sage: m = [[Q(b)(F.gen()) for b in a] for a in l for F in [K, L, M]] 

sage: d = [] 

sage: c = [] 

sage: c = [Conic(a) for a in m if a != [0,0,0]] 

sage: d = [C.has_rational_point(algorithm = 'rnfisnorm', point = True) for C in c] # long time: 3.3 seconds 

sage: all([c[k].defining_polynomial()(Sequence(d[k][1])) == 0 for k in range(len(d)) if d[k][0]]) 

True 

sage: [C.has_rational_point(algorithm='local', read_cache=False) for C in c] == [o[0] for o in d] # long time: 5 seconds 

True 

sage: [C.has_rational_point(algorithm = 'magma', read_cache=False) for C in c] == [o[0] for o in d] # long time: 3 seconds, optional - magma 

True 

 

Create a bunch of conics that are known to have rational points 

already over `\QQ` and check if points are found by 

``has_rational_point``. :: 

 

sage: l = Sequence(cartesian_product_iterator([[-1, 0, 1] for i in range(3)])) 

sage: K.<a> = QuadraticField(-23) 

sage: L.<b> = QuadraticField(19) 

sage: M.<c> = NumberField(x^5+3*x+1) 

sage: m = [[F(b) for b in a] for a in l for F in [K, L, M]] 

sage: c = [Conic(a) for a in m if a != [0,0,0] and a != [1,1,1] and a != [-1,-1,-1]] 

sage: assert all([C.has_rational_point(algorithm = 'rnfisnorm') for C in c]) 

sage: assert all([C.defining_polynomial()(Sequence(C.has_rational_point(point = True)[1])) == 0 for C in c]) 

sage: assert all([C.has_rational_point(algorithm='local', read_cache=False) for C in c]) # long time: 1 second 

""" 

if read_cache: 

if self._rational_point is not None: 

if point or obstruction: 

return True, self._rational_point 

else: 

return True 

if self._local_obstruction is not None: 

if point or obstruction: 

return False, self._local_obstruction 

else: 

return False 

if (not point) and self._finite_obstructions == [] and \ 

self._infinite_obstructions == []: 

if obstruction: 

return True, None 

return True 

if self.has_singular_point(): 

if point: 

return self.has_singular_point(point = True) 

if obstruction: 

return True, None 

return True 

B = self.base_ring() 

 

if algorithm == 'default': 

ret = self.has_rational_point(point=True, obstruction=False, 

algorithm='rnfisnorm', 

read_cache=False) 

if ret[0]: 

if point or obstruction: 

return ret 

return True 

if obstruction: 

ret = self.has_rational_point(point=False, obstruction=True, 

algorithm='local', 

read_cache=False) 

if ret[0]: 

raise RuntimeError("Outputs of algorithms in " \ 

"has_rational_point disagree " \ 

"for conic %s" % self) 

return ret 

if point: 

return False, None 

return False 

 

if algorithm == 'local': 

if point: 

raise ValueError("Algorithm 'local' cannot be combined " \ 

"with point = True in has_rational_point") 

obs = self.local_obstructions(infinite = True, finite = False, 

read_cache = read_cache) 

if obs != []: 

if obstruction: 

return False, obs[0] 

return False 

obs = self.local_obstructions(read_cache = read_cache) 

if obs == []: 

if obstruction: 

return True, None 

return True 

if obstruction: 

return False, obs[0] 

return False 

if algorithm == 'rnfisnorm': 

from sage.modules.free_module_element import vector 

if obstruction: 

raise ValueError("Algorithm rnfisnorm cannot be combined " \ 

"with obstruction = True in " \ 

"has_rational_point") 

D, T = self.diagonal_matrix() 

abc = [D[0,0], D[1,1], D[2,2]] 

for j in range(3): 

if abc[j] == 0: 

pt = self.point(T*vector({2:0,j:1})) 

if point or obstruction: 

return True, pt 

return True 

if (-abc[1]/abc[0]).is_square(): 

pt = self.point(T*vector([(-abc[1]/abc[0]).sqrt(), 1, 0])) 

if point or obstruction: 

return True, pt 

return True 

if (-abc[2]/abc[0]).is_square(): 

pt = self.point(T*vector([(-abc[2]/abc[0]).sqrt(), 0, 1])) 

if point or obstruction: 

return True, pt 

return True 

if is_RationalField(B): 

K = B 

[KtoB, BtoK] = [K.hom(K) for i in range(2)] 

else: 

K = B.absolute_field('Y') 

[KtoB, BtoK] = K.structure() 

X = PolynomialRing(K, 'X').gen() 

d = BtoK(-abc[1]/abc[0]) 

den = d.denominator() 

L = K.extension(X**2 - d*den**2, names='y') 

isnorm = BtoK(-abc[2]/abc[0]).is_norm(L, element=True) 

if isnorm[0]: 

 

pt = self.point(T*vector([KtoB(isnorm[1][0]), 

KtoB(isnorm[1][1]*den), 1])) 

if point: 

return True, pt 

return True 

if point: 

return False, None 

return False 

if algorithm == 'qfsolve': 

raise TypeError("Algorithm qfsolve in has_rational_point only " \ 

"for conics over QQ, not over %s" % B) 

if obstruction: 

raise ValueError("Invalid combination: obstruction=True and " \ 

"algorithm=%s" % algorithm) 

 

return ProjectiveConic_field.has_rational_point(self, point = point, 

algorithm = algorithm, read_cache = False) 

 

 

def is_locally_solvable(self, p): 

r""" 

Returns ``True`` if and only if ``self`` has a solution over the 

completion of the base field `B` of ``self`` at ``p``. Here ``p`` 

is a finite prime or infinite place of `B`. 

 

EXAMPLES:: 

 

sage: P.<x> = QQ[] 

sage: K.<a> = NumberField(x^3 + 5) 

sage: C = Conic(K, [1, 2, 3 - a]) 

sage: [p1, p2] = K.places() 

sage: C.is_locally_solvable(p1) 

False 

 

sage: C.is_locally_solvable(p2) 

True 

 

sage: O = K.maximal_order() 

sage: f = (2*O).factor() 

sage: C.is_locally_solvable(f[0][0]) 

True 

 

sage: C.is_locally_solvable(f[1][0]) 

False 

""" 

D, T = self.diagonal_matrix() 

abc = [D[j, j] for j in range(3)] 

for a in abc: 

if a == 0: 

return True 

a = -abc[0]/abc[2] 

b = -abc[1]/abc[2] 

 

ret = self.base_ring().hilbert_symbol(a, b, p) 

 

if ret == -1: 

if self._local_obstruction is None: 

from sage.categories.map import Map 

from sage.categories.all import Rings 

if not (isinstance(p, Map) and p.category_for().is_subcategory(Rings())) or p.codomain() is AA or p.codomain() is RLF: 

self._local_obstruction = p 

return False 

 

return True 

 

 

def local_obstructions(self, finite = True, infinite = True, read_cache = True): 

r""" 

Returns the sequence of finite primes and/or infinite places 

such that ``self`` is locally solvable at those primes and places. 

 

If the base field is `\QQ`, then the infinite place is denoted `-1`. 

 

The parameters ``finite`` and ``infinite`` (both True by default) are 

used to specify whether to look at finite and/or infinite places. 

Note that ``finite = True`` involves factorization of the determinant 

of ``self``, hence may be slow. 

 

Local obstructions are cached. The parameter ``read_cache`` 

specifies whether to look at the cache before computing anything. 

 

EXAMPLES :: 

 

sage: K.<i> = QuadraticField(-1) 

sage: Conic(K, [1, 2, 3]).local_obstructions() 

[] 

 

sage: L.<a> = QuadraticField(5) 

sage: Conic(L, [1, 2, 3]).local_obstructions() 

[Ring morphism: 

From: Number Field in a with defining polynomial x^2 - 5 

To: Algebraic Real Field 

Defn: a |--> -2.236067977499790?, Ring morphism: 

From: Number Field in a with defining polynomial x^2 - 5 

To: Algebraic Real Field 

Defn: a |--> 2.236067977499790?] 

""" 

obs0 = [] 

obs1 = [] 

B = self.base_ring() 

if infinite: 

if read_cache and self._infinite_obstructions is not None: 

obs0 = self._infinite_obstructions 

else: 

for b in B.embeddings(AA): 

if not self.is_locally_solvable(b): 

obs0.append(b) 

self._infinite_obstructions = obs0 

if finite: 

if read_cache and self._finite_obstructions is not None: 

obs1 = self._finite_obstructions 

else: 

candidates = [] 

if self.determinant() != 0: 

O = B.maximal_order() 

for a in self.symmetric_matrix().list(): 

if a != 0: 

for f in O.fractional_ideal(a).factor(): 

if f[1] < 0 and not f[0] in candidates: 

candidates.append(f[0]) 

for f in O.fractional_ideal(2*self.determinant()).factor(): 

if f[1] > 0 and not f[0] in candidates: 

candidates.append(f[0]) 

for b in candidates: 

if not self.is_locally_solvable(b): 

obs1.append(b) 

self._infinite_obstructions = obs1 

obs = obs1 + obs0 

if finite and infinite: 

assert len(obs) % 2 == 0 

return obs