def test_singularities(self):
S1 = sympify(singularities(f1,x,y))
S2 = sympify(singularities(f2,x,y))
S3 = sympify(singularities(f3,x,y))
S4 = sympify(singularities(f4,x,y))
S5 = sympify(singularities(f5,x,y))
S6 = sympify(singularities(f6,x,y))
# S7 = sympify(singularities(f7,x,y))
# S8 = sympify(singularities(f8,x,y))
S9 = sympify(singularities(f9,x,y))
S10= sympify(singularities(f10,x,y))
rt5 = sympy.roots(x**2 + 1, x).keys()
S1act = sympify([
((0,0,1),(2,2,1)),
((0,1,0),(2,1,2))
])
S2act = sympify([
((0,0,1),(3,4,2)),
((0,1,0),(4,9,1))
])
S3act = sympify([
((0,0,1),(2,1,2)),
((1,-1,1),(2,1,2)),
((1,1,1),(2,1,2))
])
S4act = sympify([
((0,0,1),(2,1,2))
])
S5act = sympify([
((0,0,1),(3,3,3)),
((1,rt5[1],0),(3,3,3)),
((1,rt5[0],0),(3,3,3))
])
S6act = sympify([
((0,0,1),(2,2,2)),
((1,0,0),(2,2,2))
])
# S7act = sympify([((0,1,0),(3,6,3))])
# S8act = sympify([
# ((0,1,0),(6,21,3)),
# ((1,0,0),(3,7,2))
# ])
S9act = sympify([((0,1,0),(5,12,1))])
S10act= sympify([
((0,1,0),(3,6,1)),
((1,0,0),(4,6,4))
])
self.assertItemsEqual(S1,S1act)
self.assertItemsEqual(S2,S2act)
self.assertItemsEqual(S3,S3act)
self.assertItemsEqual(S4,S4act)
self.assertItemsEqual(S5,S5act)
self.assertItemsEqual(S6,S6act)
# self.assertItemsEqual(S7,S7act)
# self.assertItemsEqual(S8,S8act)
self.assertItemsEqual(S9,S9act)
self.assertItemsEqual(S10,S10act)
def test_f8(self):
s = singularities(self.f8)
s_actual = [
((1, 0, 0), (5, 14, 1)),
((0, 1, 0), (2, 4, 2)),
]
self.assertEqual(s, s_actual)
def test_f2(self):
s = singularities(self.f2)
s_actual = [
((0, 0, 1), (3, 4, 2)),
((0, 1, 0), (4, 9, 1)),
]
self.assertEqual(s, s_actual)
def test_issue71(self):
R = QQ['x,y']
x,y = R.gens()
f = -x**5 + x + y**3
s = singularities(f)
s_actual = [((0,1,0),(2,2,1))]
self.assertEqual(s,s_actual)
def test_f10(self):
s = singularities(self.f10)
s_actual = [
((1,0,0),(4,6,4)),
((0,1,0),(3,6,1)),
]
self.assertEqual(s,s_actual)
def test_f10(self):
s = singularities(self.f10)
s_actual = [
((1, 0, 0), (4, 6, 4)),
((0, 1, 0), (3, 6, 1)),
]
self.assertEqual(s, s_actual)
def test_f8(self):
s = singularities(self.f8)
s_actual = [
((1,0,0),(5,14,1)),
((0,1,0),(2,4,2)),
]
self.assertEqual(s,s_actual)
def test_issue71(self):
R = QQ['x,y']
x, y = R.gens()
f = -x**5 + x + y**3
s = singularities(f)
s_actual = [((0, 1, 0), (2, 2, 1))]
self.assertEqual(s, s_actual)
def test_f6(self):
s = singularities(self.f6)
s_actual = [
((0,0,1),(2,2,2)),
((1,0,0),(2,2,2)),
]
self.assertEqual(s,s_actual)
def test_f6(self):
s = singularities(self.f6)
s_actual = [
((0, 0, 1), (2, 2, 2)),
((1, 0, 0), (2, 2, 2)),
]
self.assertEqual(s, s_actual)
def test_f2(self):
s = singularities(self.f2)
s_actual = [
((0,0,1),(3,4,2)),
((0,1,0),(4,9,1)),
]
self.assertEqual(s,s_actual)
def test_f3(self):
s = singularities(self.f3)
s_actual = [
((0,0,1),(2,1,2)),
((1,-1,1),(2,1,2)),
((1,1,1),(2,1,2)),
]
self.assertEqual(s,s_actual)
def test_f3(self):
s = singularities(self.f3)
s_actual = [
((0, 0, 1), (2, 1, 2)),
((1, -1, 1), (2, 1, 2)),
((1, 1, 1), (2, 1, 2)),
]
self.assertEqual(s, s_actual)
def differentials_numerators(f):
"""Return the numerators of a basis of holomorphic differentials on a Riemann
surface.
Parameters
----------
f : plane algebraic curve
Returns
-------
differentials : list
A list of :class:`Differential`s representing *a* basis of Abelian
differentials of the first kind.
"""
# homogenize and compute total degree
R = f.parent().change_ring(QQbar)
x,y = R.gens()
d = f.total_degree()
# construct the generalized adjoint polynomial. we want to think of it as
# an element of B[*c][x,y] where B is the base ring of f and *c are the
# indeterminates
cvars = ['c_%d_%d'%(i,j) for i in range(d-2) for j in range(d-2)]
vars = list(R.variable_names()) + cvars
C = PolynomialRing(QQbar, cvars)
S = PolynomialRing(C, [x,y])
T = PolynomialRing(QQbar, vars)
c = S.base_ring().gens()
x,y = S(x),S(y)
P = sum(c[j+(d-2)*i] * x**i * y**j
for i in range(d-2) for j in range(d-2)
if i+j <= d-3)
# for each singular point [x:y:z] = [alpha:beta:gamma], map f onto the
# "most convenient and appropriate" affine subspace, (u,v), and center at
# u=0. determine the conditions on P
singular_points = singularities(f)
conditions = []
for singular_point, _ in singular_points:
# recenter the curve and adjoint polynomial at the singular point: find
# the affine plane u,v such that the singularity occurs at u=0
g = recenter_curve(f, singular_point)
Ptilde = recenter_curve(P, singular_point)
# compute the intergral basis at the recentered singular point
# and determine the Mnuk conditions of the adjoint polynomial
b = integral_basis(g)
for bi in b:
conditions_bi = mnuk_conditions(g, bi, Ptilde)
conditions.extend(conditions_bi)
# reduce the general adjoint modulo the ideal generated by the integral
# basis conditions. the coefficients of the remaining c_ij's form the
# numerators of a basis of abelian differentials of the first kind.
#
# additionally, we try to coerce the conditions to over QQ for speed. it's
# questionable in this situation whether there is a noticible performance
# gain but it does suppress the "slow toy implementation" warning.
try:
T = T.change_ring(QQ)
ideal = T.ideal(conditions)
basis = ideal.groebner_basis()
except:
pass
ideal = T.ideal(conditions)
basis = ideal.groebner_basis()
P_reduced = P(T(x), T(y))
if basis != [0]:
P_reduced = P_reduced.reduce(basis)
U = R[S.base_ring().variable_names()]
args = [U(x),U(y)] + [U(ci) for ci in c]
Pc = P_reduced(*args)
numerators = Pc.coefficients()
return numerators
def test_f4(self):
s = singularities(self.f4)
s_actual = [((0,0,1),(2,1,2))]
self.assertEqual(s,s_actual)
def differentials_numerators(f):
"""Return the numerators of a basis of holomorphic differentials on a Riemann
surface.
Parameters
----------
f : plane algebraic curve
Returns
-------
differentials : list
A list of :class:`Differential`s representing *a* basis of Abelian
differentials of the first kind.
"""
# homogenize and compute total degree
R = f.parent().change_ring(QQbar)
x,y = R.gens()
d = f.total_degree()
# construct the generalized adjoint polynomial. we want to think of it as
# an element of B[*c][x,y] where B is the base ring of f and *c are the
# indeterminates
cvars = ['c_%d_%d'%(i,j) for i in range(d-2) for j in range(d-2)]
vars = list(R.variable_names()) + cvars
C = PolynomialRing(QQbar, cvars)
S = PolynomialRing(C, [x,y])
T = PolynomialRing(QQbar, vars)
c = S.base_ring().gens()
x,y = S(x),S(y)
P = sum(c[j+(d-2)*i] * x**i * y**j
for i in range(d-2) for j in range(d-2)
if i+j <= d-3)
# for each singular point [x:y:z] = [alpha:beta:gamma], map f onto the
# "most convenient and appropriate" affine subspace, (u,v), and center at
# u=0. determine the conditions on P
singular_points = singularities(f)
conditions = []
for singular_point, _ in singular_points:
# recenter the curve and adjoint polynomial at the singular point: find
# the affine plane u,v such that the singularity occurs at u=0
g = recenter_curve(f, singular_point)
Ptilde = recenter_curve(P, singular_point)
# compute the intergral basis at the recentered singular point
# and determine the Mnuk conditions of the adjoint polynomial
b = integral_basis(g)
for bi in b:
conditions_bi = mnuk_conditions(g, bi, Ptilde)
conditions.extend(conditions_bi)
# reduce the general adjoint modulo the ideal generated by the integral
# basis conditions. the coefficients of the remaining c_ij's form the
# numerators of a basis of abelian differentials of the first kind.
#
# additionally, we try to coerce the conditions to over QQ for speed. it's
# questionable in this situation whether there is a noticible performance
# gain but it does suppress the "slow toy implementation" warning.
try:
T = T.change_ring(QQ)
ideal = T.ideal(conditions)
basis = ideal.groebner_basis()
except:
pass
ideal = T.ideal(conditions)
basis = ideal.groebner_basis()
P_reduced = P(T(x), T(y))
if basis != [0]:
P_reduced = P_reduced.reduce(basis)
U = R[S.base_ring().variable_names()]
args = [U(x),U(y)] + [U(ci) for ci in c]
Pc = P_reduced(*args)
numerators = Pc.coefficients()
return numerators
def test_f7(self):
s = singularities(self.f7)
s_actual = [((0,1,0),(3,6,3))]
self.assertEqual(s,s_actual)
def test_f9(self):
s = singularities(self.f9)
s_actual = [((0,1,0),(5,12,1))]
self.assertEqual(s,s_actual)
def test_f9(self):
s = singularities(self.f9)
s_actual = [((0, 1, 0), (5, 12, 1))]
self.assertEqual(s, s_actual)
def test_f4(self):
s = singularities(self.f4)
s_actual = [((0, 0, 1), (2, 1, 2))]
self.assertEqual(s, s_actual)
def test_f7(self):
s = singularities(self.f7)
s_actual = [((0, 1, 0), (3, 6, 3))]
self.assertEqual(s, s_actual)
