def test_rcvexample(self):
R = QQ['x,y']
x,y = R.gens()
f = x**2*y**3 - x**4 + 1
a = integral_basis(f)
b = [1, x*y, x**2*y**2]
self.assertEqual(a,b)
def test_rcvexample_monicized(self):
R = QQ['x,y']
x,y = R.gens()
f = y**3 - x**8 + x**4
a = integral_basis(f)
b = [1, y/x, y**2/x**2]
self.assertEqual(a,b)
def test_issue86(self):
R = QQ['x,z']
x,z = R.gens()
f = -x**7 + 2*x*z**5 + z**4
a = integral_basis(f)
b = [1, 2*x*z, 2*z*(2*x*z + 1)/x, 4*z**2*(2*x*z + 1)/x**3,
8*z**3*(2*x*z + 1)/x**5]
self.assertEqual(a,b)
def test_f8b(self):
# the curve f8 recentered at the singular point (0 : 1 : 0)
R = QQ['x,y']
x,y = R.gens()
g = -y**8 + 2*x**3 + x**2
a = integral_basis(g)
b = [1, y, y**2, y**3, y**4/x, y**5/x, y**6/x, y**7/x]
self.assertEqual(a,b)
def test_integral_basis(self):
self.assertEqual(integral_basis(f1,x,y),
[1, y*(x**2 - x + 1)/x**2])
self.assertEqual(integral_basis(f2,x,y),
[1, y/x, y**2/x**3])
self.assertEqual(integral_basis(f4,x,y),
[1, y/x])
# # long time
# self.assertEqual(integral_basis(f5,x,y),
# [])
# # long time
# self.assertEqual(integral_basis(f6,x,y),
# [1, y, y**2/x, y**3/x])
self.assertEqual(integral_basis(f7,x,y),
[1, y, y**2])
# self.assertEqual(integral_basis(f8,x,y),
# [])
self.assertEqual(integral_basis(f9,x,y),
[1, y, y**2])
self.assertEqual(integral_basis(f10,x,y),
[1, x*y, x**2*y**2, x**3*y**3])
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 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_f10(self):
x,y = self.f10.parent().gens()
a = integral_basis(self.f10)
b = [1, x*y, x**2*y**2, x**3*y**3]
self.assertEqual(a,b)
def test_f9(self):
x,y = self.f9.parent().gens()
a = integral_basis(self.f9)
b = [1, y, y**2]
self.assertEqual(a,b)
def test_f8(self):
x,y = self.f8.parent().gens()
a = integral_basis(self.f8)
b = [1, x*y, x*y**2, y**3*x, x**2*y**4, y**5*x**2]
self.assertEqual(a,b)
def test_f6(self):
x,y = self.f6.parent().gens()
a = integral_basis(self.f6)
b = [1, y, y**2/x, y**3/x]
self.assertEqual(a,b)
def test_f5(self):
x,y = self.f5.parent().gens()
a = integral_basis(self.f5)
b = [1, y, y**2, y**3, y*(y**3-1)/x, y**2*(y**3-1)/x**2]
self.assertEqual(a,b)
def test_f4(self):
x,y = self.f4.parent().gens()
a = integral_basis(self.f4)
b = [1, y/x]
self.assertEqual(a,b)
def test_f3(self):
x,y = self.f3.parent().gens()
a = integral_basis(self.f3)
b = [1, y, (y**2-1)/(x-1), -y*(x - 4*y**2 + 3)/(x*(x - 1))]
self.assertEqual(a,b)
def test_f2(self):
x,y = self.f2.parent().gens()
a = integral_basis(self.f2)
b = [1, y/x, y**2/x**3]
self.assertEqual(a,b)
def test_f1(self):
x,y = self.f1.parent().gens()
a = integral_basis(self.f1)
b = [1, y*(x**2-x+1)/x**2]
self.assertEqual(a,b)