def test_f2(self):
x, y = self.f2.parent().gens()
dfdy = self.f2.derivative(y)
X = DummyRS(self.f2)
a = map(lambda omega: omega.as_expression(), differentials(X))
b = [x * y / dfdy, x**3 / dfdy]
self.assertEqual(a, b)
def test_hyperelliptic_regular_places(self):
R = QQ['x,y']
x, y = R.gens()
X = RiemannSurface(y**2 - (x + 1) * (x - 1) * (x - 2) * (x + 2))
omegas = differentials(X)
# the places above x=0 are regular
places = X(0)
for P in places:
a, b = P.x, P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a, b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
# the places above x=oo are regular: P = (1/t, \pm 1/t**2 + O(1))
# (however, all places at infinity are treated as discriminant)
#
# in this particular example, omega[0] = 1/(2*y). At the places at
# infinity, these are equal to \mp 0.5, respectively. (the switch in
# sign comes from the derivative dxdt = -1/t**2)
places = X('oo')
for P in places:
sign = P.puiseux_series.ypart[-2]
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = -sign * 0.5
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
def test_f1(self):
x, y = self.f1.parent().gens()
dfdy = self.f1.derivative(y)
X = DummyRS(self.f1)
a = map(lambda omega: omega.as_expression(), differentials(X))
b = []
self.assertEqual(a, b)
def test_f1(self):
x,y = self.f1.parent().gens()
dfdy = self.f1.derivative(y)
X = DummyRS(self.f1)
a = map(lambda omega: omega.as_expression(), differentials(X))
b = []
self.assertEqual(a,b)
def test_f2(self):
x,y = self.f2.parent().gens()
dfdy = self.f2.derivative(y)
X = DummyRS(self.f2)
a = map(lambda omega: omega.as_expression(), differentials(X))
b = [x*y/dfdy, x**3/dfdy]
self.assertEqual(a,b)
def test_hyperelliptic_regular_places(self):
R = QQ['x,y']
x,y = R.gens()
X = RiemannSurface(y**2 - (x+1)*(x-1)*(x-2)*(x+2))
omegas = differentials(X)
# the places above x=0 are regular
places = X(0)
for P in places:
a,b = P.x,P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a,b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1-val2), 1e-8)
# the places above x=oo are regular: P = (1/t, \pm 1/t**2 + O(1))
# (however, all places at infinity are treated as discriminant)
#
# in this particular example, omega[0] = 1/(2*y). At the places at
# infinity, these are equal to \mp 0.5, respectively. (the switch in
# sign comes from the derivative dxdt = -1/t**2)
places = X('oo')
for P in places:
sign = P.puiseux_series.ypart[-2]
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = -sign*0.5
val2 = omega_P(CC(0))
self.assertLess(abs(val1-val2), 1e-8)
def test_f2(self):
x, y = self.f2.parent().gens()
dfdy = self.f2.derivative(y)
X = DummyRS(self.f2)
a = [omega.as_expression() for omega in differentials(X)]
b = [x * y / dfdy, x**3 / dfdy]
self.assertEqual(a, b)
def test_f1(self):
x, y = self.f1.parent().gens()
dfdy = self.f1.derivative(y)
X = DummyRS(self.f1)
a = [omega.as_expression() for omega in differentials(X)]
b = []
self.assertEqual(a, b)
def test_differentials(self):
self.assertItemsEqual(
differentials(f1,x,y),
[],
)
self.assertItemsEqual(
differentials(f2,x,y),
[x**3/(2*x**3 + 3*y**2), x*y/(2*x**3 + 3*y**2)],
)
self.assertItemsEqual(
differentials(f3,x,y),
[],
)
self.assertItemsEqual(
differentials(f4,x,y),
[],
)
def test_f2_regular_places(self):
X = self.X2
omegas = differentials(X)
# the places above x=1 are regular
places = X(1)
for P in places:
a, b = P.x, P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a, b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
def test_f2_regular_places(self):
X = self.X2
omegas = differentials(X)
# the places above x=1 are regular
places = X(1)
for P in places:
a,b = P.x,P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a,b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1-val2), 1e-8)
def holomorphic_differentials(self):
r"""Returns a basis of holomorphic differentials on the surface.
Parameters
----------
None
Returns
-------
differentials : list
A list of holomorphic differentials forming a basis.
"""
value = differentials(self)
return value
