def create_key_and_extra_args(self,
gens,
check_integral=True,
check_rank=True,
check_is_ring=True,
is_maximal=None,
allow_subfield=False):
if allow_subfield:
raise NotImplementedError(
"the allow_subfield parameter is not supported yet")
if len(gens) == 0:
raise ValueError("gens must span an order over ZZ")
from sage.all import Sequence
gens = Sequence(gens)
K = gens.universe()
from sage.rings.number_field.order import is_NumberFieldOrder
if is_NumberFieldOrder(K):
K = K.number_field()
gens = frozenset([K(g) for g in gens])
return (K, gens), {
"check_integral": check_integral,
"check_rank": check_rank,
"check_is_ring": check_is_ring,
"is_maximal": is_maximal
}
def global_height(self, prec=None):
r"""
Returns the logarithmic height of the point.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number
EXAMPLES::
sage: P.<x,y>=AffineSpace(QQ,2)
sage: Q=P(41,1/12)
sage: Q.global_height()
3.71357206670431
::
sage: P=AffineSpace(ZZ,4,'x')
sage: Q=P(3,17,-51,5)
sage: Q.global_height()
3.93182563272433
::
sage: R.<x>=PolynomialRing(QQ)
sage: k.<w>=NumberField(x^2+5)
sage: A=AffineSpace(k,2,'z')
sage: A([3,5*w+1]).global_height(prec=100)
2.4181409534757389986565376694
.. TODO::
p-adic heights
add heights to integer.pyx and remove special case
"""
if self.domain().base_ring() == ZZ:
if prec is None:
R = RealField()
else:
R = RealField(prec)
H = max([
self[i].abs() for i in range(
self.codomain().ambient_space().dimension_relative())
])
return (R(max(H, 1)).log())
if self.domain().base_ring() in _NumberFields or is_NumberFieldOrder(
self.domain().base_ring()):
return (max([
self[i].global_height(prec) for i in range(
self.codomain().ambient_space().dimension_relative())
]))
else:
raise NotImplementedError(
"Must be over a Numberfield or a Numberfield Order")
def global_height(self, prec=None):
r"""
Returns the absolute logarithmic height of the point.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: Q = P.point([4, 4, 1/30])
sage: Q.global_height()
4.78749174278205
::
sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
sage: Q = P([4, 1, 30])
sage: Q.global_height()
3.40119738166216
::
sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2+5)
sage: A = ProjectiveSpace(k, 2, 'z')
sage: A([3, 5*w+1, 1]).global_height(prec=100)
2.4181409534757389986565376694
::
sage: P.<x,y,z> = ProjectiveSpace(QQbar,2)
sage: Q = P([QQbar(sqrt(3)), QQbar(sqrt(-2)), 1])
sage: Q.global_height()
0.549306144334055
::
sage: K = UniversalCyclotomicField()
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: Q = P.point([K(4/3), K.gen(7), K.gen(5)])
sage: Q.global_height()
1.38629436111989
"""
K = self.codomain().base_ring()
if K in _NumberFields or is_NumberFieldOrder(K):
P = self
else:
try:
P = self._number_field_from_algebraics()
except TypeError:
raise TypeError("must be defined over an algebraic field")
return(max([P[i].global_height(prec=prec) for i in range(self.codomain().ambient_space().dimension_relative()+1)]))
def global_height(self, prec=None):
r"""
Return the absolute logarithmic height of the point.
This function computes the maximum of global height of each
component point in the product. Global height of component
point is computed using function for projective point.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number.
EXAMPLES::
sage: PP = ProductProjectiveSpaces(QQ, [2,2], 'x')
sage: Q = PP([1, 7, 5, 18, 2, 3])
sage: Q.global_height()
1.94591014905531
::
sage: PP = ProductProjectiveSpaces(ZZ, [1,1], 'x')
sage: A = PP([-30, 2, 1, 6])
sage: A.global_height()
3.40119738166216
::
sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2 + 5)
sage: PP = ProductProjectiveSpaces(k, [1, 2], 'y')
sage: Q = PP([3, 5*w+1, 1, 7*w, 10])
sage: Q.global_height()
2.30258509299405
::
sage: PP = ProductProjectiveSpaces(QQbar, [1, 1], 'x')
sage: Q = PP([1, QQbar(sqrt(2)), QQbar(5^(1/3)), QQbar(3^(1/3))])
sage: Q.global_height()
0.536479304144700
"""
K = self.codomain().base_ring()
if K not in NumberFields(
) and not is_NumberFieldOrder(K) and K != QQbar:
raise TypeError(
"must be over a number field or a number field order or QQbar")
n = self.codomain().ambient_space().num_components()
return max(self[i].global_height(prec=prec) for i in range(n))
def global_height(self, prec=None):
r"""
Returns the logarithmic height of the point.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number
EXAMPLES::
sage: P.<x,y>=AffineSpace(QQ,2)
sage: Q=P(41,1/12)
sage: Q.global_height()
3.71357206670431
::
sage: P=AffineSpace(ZZ,4,'x')
sage: Q=P(3,17,-51,5)
sage: Q.global_height()
3.93182563272433
::
sage: R.<x>=PolynomialRing(QQ)
sage: k.<w>=NumberField(x^2+5)
sage: A=AffineSpace(k,2,'z')
sage: A([3,5*w+1]).global_height(prec=100)
2.4181409534757389986565376694
.. TODO::
p-adic heights
add heights to integer.pyx and remove special case
"""
if self.domain().base_ring() == ZZ:
if prec is None:
R = RealField()
else:
R = RealField(prec)
H = max([self[i].abs() for i in range(self.codomain().ambient_space().dimension_relative())])
return R(max(H, 1)).log()
if self.domain().base_ring() in _NumberFields or is_NumberFieldOrder(self.domain().base_ring()):
return max(
[self[i].global_height(prec) for i in range(self.codomain().ambient_space().dimension_relative())]
)
else:
raise NotImplementedError("Must be over a Numberfield or a Numberfield Order")
def global_height(self, prec=None):
r"""
Returns the logarithmic height of the points. Must be over `\ZZ` or `\QQ`.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number
EXAMPLES::
sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: Q=P.point([4,4,1/30])
sage: Q.global_height()
4.78749174278205
::
sage: P.<x,y,z>=ProjectiveSpace(ZZ,2)
sage: Q=P([4,1,30])
sage: Q.global_height()
3.40119738166216
::
sage: R.<x>=PolynomialRing(QQ)
sage: k.<w>=NumberField(x^2+5)
sage: A=ProjectiveSpace(k,2,'z')
sage: A([3,5*w+1,1]).global_height(prec=100)
2.4181409534757389986565376694
.. TODO::
p-adic heights
add heights to integer.pyx and remove special case
"""
if self.domain().base_ring() == ZZ:
if prec is None:
R = RealField()
else:
R = RealField(prec)
H=R(0)
return(R(max([self[i].abs() for i in range(self.codomain().ambient_space().dimension_relative()+1)])).log())
if self.domain().base_ring() in _NumberFields or is_NumberFieldOrder(self.domain().base_ring()):
return(max([self[i].global_height(prec) for i in range(self.codomain().ambient_space().dimension_relative()+1)]))
else:
raise NotImplementedError("Must be over a Numberfield or a Numberfield Order")
def global_height(self, prec=None):
r"""
Returns the maximum of the absolute logarithmic heights of the coefficients
in any of the coordinate functions of this map.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number.
.. TODO::
Add functionality for `\QQbar`, implement function to convert
the map defined over `\QQbar` to map over a number field.
EXAMPLES::
sage: P1xP1.<x,y,u,v> = ProductProjectiveSpaces([1, 1], ZZ)
sage: H = End(P1xP1)
sage: f = H([x^2*u, 3*y^2*v, 5*x*v^2, y*u^2])
sage: f.global_height()
1.60943791243410
::
sage: u = QQ['u'].0
sage: R = NumberField(u^2 - 2, 'v')
sage: PP.<x,y,a,b> = ProductProjectiveSpaces([1, 1], R)
sage: H = End(PP)
sage: O = R.maximal_order()
sage: g = H([3*O(u)*x^2, 13*x*y, 7*a*y, 5*b*x + O(u)*a*y])
sage: g.global_height()
2.56494935746154
"""
K = self.domain().base_ring()
if K in NumberFields() or is_NumberFieldOrder(K):
H = 0
for i in range(self.domain().ambient_space().ngens()):
C = self[i].coefficients()
h = max(c.global_height(prec=prec) for c in C)
H = max(H, h)
return H
elif K == QQbar:
raise NotImplementedError("not implemented for QQbar")
else:
raise TypeError(
"Must be over a Numberfield or a Numberfield Order or QQbar")
def global_height(self, prec=None):
r"""
Returns the logarithmic height of the points.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number
EXAMPLES::
sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: Q=P.point([4,4,1/30])
sage: Q.global_height()
4.78749174278205
::
sage: P.<x,y,z>=ProjectiveSpace(ZZ,2)
sage: Q=P([4,1,30])
sage: Q.global_height()
3.40119738166216
::
sage: R.<x>=PolynomialRing(QQ)
sage: k.<w>=NumberField(x^2+5)
sage: A=ProjectiveSpace(k,2,'z')
sage: A([3,5*w+1,1]).global_height(prec=100)
2.4181409534757389986565376694
"""
if self.domain().base_ring() in _NumberFields or is_NumberFieldOrder(self.domain().base_ring()):
return(max([self[i].global_height(prec=prec) for i in range(self.codomain().ambient_space().dimension_relative()+1)]))
else:
raise TypeError("Must be over a Numberfield or a Numberfield Order")
def global_height(self, prec=None):
r"""
Return the absolute logarithmic height of the point.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT:
- a real number.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: Q = P.point([4, 4, 1/30])
sage: Q.global_height()
4.78749174278205
::
sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
sage: Q = P([4, 1, 30])
sage: Q.global_height()
3.40119738166216
::
sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2+5)
sage: A = ProjectiveSpace(k, 2, 'z')
sage: A([3, 5*w+1, 1]).global_height(prec=100)
2.4181409534757389986565376694
::
sage: P.<x,y,z> = ProjectiveSpace(QQbar,2)
sage: Q = P([QQbar(sqrt(3)), QQbar(sqrt(-2)), 1])
sage: Q.global_height()
0.549306144334055
::
sage: K = UniversalCyclotomicField()
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: Q = P.point([K(4/3), K.gen(7), K.gen(5)])
sage: Q.global_height()
1.38629436111989
TESTS::
sage: P = ProjectiveSpace(QQ, 2)
sage: P(1/1,2/3,5/8).global_height()
3.17805383034795
sage: x = polygen(QQ, 'x')
sage: F.<u> = NumberField(x^3 - 5)
sage: P = ProjectiveSpace(F, 2)
sage: P(u,u^2/5,1).global_height()
1.07295860828940
"""
if prec is None:
prec = 53
K = self.codomain().base_ring()
if K in _NumberFields or is_NumberFieldOrder(K):
P = self
else:
try:
P = self._number_field_from_algebraics()
except TypeError:
raise TypeError("must be defined over an algebraic field")
else:
K = P.codomain().base_ring()
# first get rid of the denominators
denom = lcm([xi.denominator() for xi in P])
x = [xi * denom for xi in P]
d = K.degree()
if d == 1:
height = max(abs(xi) for xi in x) / gcd(x)
return height.log().n(prec=prec)
finite = ~sum(K.ideal(xi) for xi in x).norm()
infinite = prod(
max(abs(xi.complex_embedding(prec, i)) for xi in x)
for i in range(d))
height = (finite * infinite)**(~d)
return height.log()