def setUp(self):
R = QQ['x,y']
x, y = R.gens()
f1 = y**2 - x
self.f1 = f1
self.X1 = RiemannSurface(f1)
f2 = y**3 - x
self.f2 = f2
self.X2 = RiemannSurface(f2)
def setUp(self):
R = QQ['x,y']
x,y = R.gens()
f1 = y**2 - x
self.f1 = f1
self.X1 = RiemannSurface(f1)
f2 = y**2 - (x**2 + 1)
self.f2 = f2
self.X2 = RiemannSurface(f2)
f3 = y**2 - (x**4 - 1)
self.f3 = f3
self.X3 = RiemannSurface(f3)
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_regular_places(self):
X = RiemannSurface(self.f1)
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_from_cycle_issue(self):
# there is an unlabeled issue with the construction of certain cycles
# on the curve f2. it turned out that RiemannSurfacePath_from_cycle had
# incorrectly created a ComplexPath from the segments
#
# this tests is here just to assert that no errors are raised or that
# the calculation doesn't time out
R = QQ['x,y']
x,y = R.gens()
f = -x**7 + 2*x**3*y + y**3
X = RiemannSurface(f)
PF = X.path_factory
a_tuples = PF.skeleton.a_cycles()
b_tuples = PF.skeleton.b_cycles()
for cycle in a_tuples + b_tuples:
gamma = PF.RiemannSurfacePath_from_cycle(cycle)
def setUp(self):
R = QQ['x,y']; x,y = R.gens()
f43 = y**3*((y**4-16)**2 - 2*x) - x**12
self.f43 = f43
self.X43 = RiemannSurface(f43)
_ = self.X43.discriminant_points
f1 = (x**2 - x + 1)*y**2 - 2*x**2*y + x**4
f2 = -x**7 + 2*x**3*y + y**3
f3 = (y**2-x**2)*(x-1)*(2*x-3) - 4*(x**2+y**2-2*x)**2
f4 = y**2 + x**3 - x**2
f5 = (x**2 + y**2)**3 + 3*x**2*y - y**3
f6 = y**4 - y**2*x + x**2
f7 = y**3 - (x**3 + y)**2 + 1
f8 = (x**6)*y**3 + 2*x**3*y - 1
f9 = 2*x**7*y + 2*x**7 + y**3 + 3*y**2 + 3*y
f10 = (x**3)*y**4 + 4*x**2*y**2 + 2*x**3*y - 1
f11 = y**2 - (x-2)*(x-1)*(x+1)*(x+2)
f12 = x**4 + y**4 - 1
f = f2
X = RiemannSurface(f,x,y)
print '=== base point ==='
print X.base_point()
print '\n=== base sheets ==='
print X.base_sheets()
PF = X.PF
# print '\n=== plotting x-path ==='
# t = linspace(0,1,32)
# gamma.plot_x(t)
print '\n=== branch_points ==='
b = PF.branch_points()
print b