def xseries(self, all_conjugates=True):
r"""Returns the corresponding x-series.
Parameters
----------
all_conjugates : bool
(default: True) If ``True``, returns all conjugates
x-representations of this Puiseux t-series. If ``False``,
only returns one representative.
Returns
-------
list
List of PuiseuxXSeries representations of this PuiseuxTSeries.
"""
# obtain relevant rings:
# o R = parent ring of curve
# o L = parent ring of T-series
# o S = temporary polynomial ring over base ring of T-series
# o P = Puiseux series ring
L = self.ypart.parent()
t = L.gen()
S = L.base_ring()['z']
z = S.gen()
R = self.f.parent()
x,y = R.gens()
P = PuiseuxSeriesRing(L.base_ring(), str(x))
x = P.gen()
# given x = alpha + lambda*t^e solve for t. this involves finding an
# e-th root of either (1/lambda) or of lambda, depending on e's sign
## (A sign on a ramification index ? hm)
e = self.ramification_index
abse = abs(e)
lamb = S(self.xcoefficient)
order = self.order
if e > 0:
phi = lamb*z**e - 1
else:
phi = z**abse - lamb
mu = phi.roots(QQbar, multiplicities=False)[0]
if all_conjugates:
zeta_e=QQbar.zeta(abse)
conjugates = [mu*zeta_e**k for k in range(abse)]
else:
conjugates = [mu]
map(lambda x: x.exactify(), conjugates)
# determine the resulting x-series
xseries = []
for c in conjugates:
t = self.ypart.parent().gen()
fconj = self.ypart(c*t)
p = P(fconj(x**(QQ(1)/e)))
p = p.add_bigoh(QQ(order+1)/abse)
xseries.append(p)
return xseries
def xseries(self, all_conjugates=True):
r"""Returns the corresponding x-series.
Parameters
----------
all_conjugates : bool
(default: True) If ``True``, returns all conjugates
x-representations of this Puiseux t-series. If ``False``,
only returns one representative.
Returns
-------
list
List of PuiseuxXSeries representations of this PuiseuxTSeries.
"""
# obtain relevant rings:
# o R = parent ring of curve
# o L = parent ring of T-series
# o S = temporary polynomial ring over base ring of T-series
# o P = Puiseux series ring
L = self.ypart.parent()
t = L.gen()
S = L.base_ring()['z']
z = S.gen()
R = self.f.parent()
x,y = R.gens()
P = PuiseuxSeriesRing(L.base_ring(), str(x))
x = P.gen()
# given x = alpha + lambda*t^e solve for t. this involves finding an
# e-th root of either (1/lambda) or of lambda, depending on e's sign
## (A sign on a ramification index ? hm)
e = self.ramification_index
abse = abs(e)
lamb = S(self.xcoefficient)
order = self.order
if e > 0:
phi = lamb*z**e - 1
else:
phi = z**abse - lamb
mu = phi.roots(QQbar, multiplicities=False)[0]
if all_conjugates:
zeta_e=QQbar.zeta(abse)
conjugates = [mu*zeta_e**k for k in range(abse)]
else:
conjugates = [mu]
map(lambda x: x.exactify(), conjugates)
# determine the resulting x-series
xseries = []
for c in conjugates:
t = self.ypart.parent().gen()
fconj = self.ypart(c*t)
p = P(fconj(x**(QQ(1)/e)))
p = p.add_bigoh(QQ(order+1)/abse)
xseries.append(p)
return xseries
def compute_bd(f, b, df, r, alpha):
"""Determine the next integral basis element form those already computed."""
# obtain the ring of Puiseux series in which the truncated series
# live. these should already be such that the base ring is SR, the symbolic
# ring. (below we will have to introduce symbolic indeterminants)
R = f.parent()
F = R.fraction_field()
x, y = R.gens()
# construct a list of indeterminants and a guess for the next integral
# basis element. to make computations uniform in the univariate and
# multivariate cases an additional generator of the underlying polynomial
# ring is introduced.
d = len(b)
Q = PolynomialRing(QQbar, ['a%d' % n for n in range(d)] + ['dummy'])
a = tuple(Q.gens())
b = tuple(b)
P = PuiseuxSeriesRing(Q, str(x))
xx = P.gen()
bd = F(y * b[-1])
# XXX HACK
for l in range(len(r)):
for k in range(len(r[l])):
r[l][k] = r[l][k].change_ring(Q)
# sufficiently singularize the current integral basis element guess at each
# of the singular points of df
for l in range(len(df)):
k = df[l] # factor
alphak = alpha[l] # point at which the truncated series are centered
rk = r[l] # truncated puiseux series
# singularize the current guess at the current point using each
# truncated Puiseux seriesx
sufficiently_singular = False
while not sufficiently_singular:
# from each puiseux series, rki, centered at alphak construct a
# system of equations from the negative exponent terms appearing in
# the expression A(x,rki))
equations = []
for rki in rk:
# A = sum(a[j] * b[j](xx,rki) for j in range(d))
A = evaluate_A(a, b, xx, rki, d)
A += bd(xx, rki)
# implicit division by x-alphak, hence truncation to x^1
terms = A.truncate(1).coefficients()
equations.extend(terms)
# attempt to solve this linear system of equations. if a (unique)
# solution exists then the integral basis element is not singular
# enough at alphak
sols = solve_coefficient_system(Q, equations, a)
if not sols is None:
bdm1 = sum(F(sols[i][0]) * b[i] for i in range(d))
bd = F(bdm1 + bd) / F(k)
else:
sufficiently_singular = True
return bd
def compute_bd(f, b, df, r, alpha):
"""Determine the next integral basis element form those already computed."""
# obtain the ring of Puiseux series in which the truncated series
# live. these should already be such that the base ring is SR, the symbolic
# ring. (below we will have to introduce symbolic indeterminants)
R = f.parent()
F = R.fraction_field()
x,y = R.gens()
# construct a list of indeterminants and a guess for the next integral
# basis element. to make computations uniform in the univariate and
# multivariate cases an additional generator of the underlying polynomial
# ring is introduced.
d = len(b)
Q = PolynomialRing(QQbar, ['a%d'%n for n in range(d)] + ['dummy'])
a = tuple(Q.gens())
b = tuple(b)
P = PuiseuxSeriesRing(Q, str(x))
xx = P.gen()
bd = F(y*b[-1])
# XXX HACK
for l in range(len(r)):
for k in range(len(r[l])):
r[l][k] = r[l][k].change_ring(Q)
# sufficiently singularize the current integral basis element guess at each
# of the singular points of df
for l in range(len(df)):
k = df[l] # factor
alphak = alpha[l] # point at which the truncated series are centered
rk = r[l] # truncated puiseux series
# singularize the current guess at the current point using each
# truncated Puiseux seriesx
sufficiently_singular = False
while not sufficiently_singular:
# from each puiseux series, rki, centered at alphak construct a
# system of equations from the negative exponent terms appearing in
# the expression A(x,rki))
equations = []
for rki in rk:
# A = sum(a[j] * b[j](xx,rki) for j in range(d))
A = evaluate_A(a,b,xx,rki,d)
A += bd(xx, rki)
# implicit division by x-alphak, hence truncation to x^1
terms = A.truncate(1).coefficients()
equations.extend(terms)
# attempt to solve this linear system of equations. if a (unique)
# solution exists then the integral basis element is not singular
# enough at alphak
sols = solve_coefficient_system(Q, equations, a)
if not sols is None:
bdm1 = sum(F(sols[i][0])*b[i] for i in range(d))
bd = F(bdm1 + bd)/ F(k)
else:
sufficiently_singular = True
return bd
def test_change_ring(self):
R = PuiseuxSeriesRing(QQ, 'x')
S = R.change_ring(QQbar)
self.assertEqual(R.base_ring(), QQ)
self.assertEqual(S.base_ring(), QQbar)
T = R.change_ring(SR)
self.assertEqual(T.base_ring(), SR)
B = QQ['a,b']
U = R.change_ring(B)
self.assertEqual(U.base_ring(), B)
def test_sub(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1)/QQ(2)
p = 1 + t
q = 1 + t + t**2
r = t**2
self.assertEqual(q - p, r)
p = 1 + t**half
q = 1 + t**half + t
r = t
self.assertEqual(q - p, r)
def test_sub(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1) / QQ(2)
p = 1 + t
q = 1 + t + t**2
r = t**2
self.assertEqual(q - p, r)
p = 1 + t**half
q = 1 + t**half + t
r = t
self.assertEqual(q - p, r)
def test_bigoh(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
half = QQ(1)/QQ(2)
p = x**(3*half)
q = p.add_bigoh(half)
self.assertEqual(q.prec(), half)
self.assertEqual(q.laurent_part.prec(), 1)
p = x**(3*half)
q = p.add_bigoh(4)
self.assertEqual(q.prec(), 4)
self.assertEqual(q.laurent_part.prec(), 8)
def test_bigoh(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
half = QQ(1) / QQ(2)
p = x**(3 * half)
q = p.add_bigoh(half)
self.assertEqual(q.prec(), half)
self.assertEqual(q.laurent_part.prec(), 1)
p = x**(3 * half)
q = p.add_bigoh(4)
self.assertEqual(q.prec(), 4)
self.assertEqual(q.laurent_part.prec(), 8)
def test_change_ring(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
half = QQ(1)/2
p = x**(-half) + 1 + x + half*x**(5*half)
S = PuiseuxSeriesRing(CC, 'x')
q = p.change_ring(CC)
self.assertEqual(q.parent(), S)
T = PuiseuxSeriesRing(SR, 'x')
r = p.change_ring(SR)
self.assertEqual(r.parent(), T)
B = QQ['a,b']
U = PuiseuxSeriesRing(B, 'x')
s = p.change_ring(B)
self.assertEqual(s.parent(), U)
def test_mul(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1) / QQ(2)
p = t**half
q = t**half
r = t
self.assertEqual(p * q, r)
p = 1 + t
q = 1 + t + t**2
r = 1 + 2 * t + 2 * t**2 + t**3
self.assertEqual(p * q, r)
p = 1 + t**half
q = 1 + t**half + t
r = 1 + 2 * t**half + 2 * t + t**(half + 1)
self.assertEqual(p * q, r)
def test_mul(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1)/QQ(2)
p = t**half
q = t**half
r = t
self.assertEqual(p * q, r)
p = 1 + t
q = 1 + t + t**2
r = 1 + 2*t + 2*t**2 + t**3
self.assertEqual(p * q, r)
p = 1 + t**half
q = 1 + t**half + t
r = 1 + 2*t**half + 2*t + t**(half+1)
self.assertEqual(p * q, r)
def test_repr(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
p = R(1)
s = '1'
self.assertEqual(str(p), s)
p = t
s = 't'
self.assertEqual(str(p), s)
p = t**2
s = 't^2'
self.assertEqual(str(p), s)
half = QQ(1) / QQ(2)
p = t**half
s = 't^(1/2)'
self.assertEqual(str(p), s)
p = t**(-half)
s = 't^(-1/2)'
self.assertEqual(str(p), s)
def test_laurent_ramification(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
y = R.laurent_series_ring().gen()
p = x
self.assertEqual(p.laurent_part, y)
self.assertEqual(p.ramification_index, 1)
p = x**2
self.assertEqual(p.laurent_part, y**2)
self.assertEqual(p.ramification_index, 1)
p = x**(QQ(1) / 2)
self.assertEqual(p.laurent_part, y)
self.assertEqual(p.ramification_index, 2)
p = x**(QQ(2) / 3)
self.assertEqual(p.laurent_part, y**2)
self.assertEqual(p.ramification_index, 3)
p = 1 + 42 * x**(QQ(1) / 2) + 99 * x**(QQ(1) / 3)
self.assertEqual(p.laurent_part, 1 + 99 * y**2 + 42 * y**3)
self.assertEqual(p.ramification_index, 6)
def test_add(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1)/QQ(2)
p = 1
q = t
r = 1 + t
self.assertEqual(p + q, r)
p = 1
q = t**half
r = 1 + t**half
self.assertEqual(p + q, r)
p = 1 + t
q = 1 + t + t**2
r = 2 + 2*t + t**2
self.assertEqual(p + q, r)
p = 1 + t**(QQ(1)/2)
q = 1 + t**(QQ(1)/2) + t
r = 2 + 2*t**(QQ(1)/2) + t
self.assertEqual(p + q, r)
def test_laurent_ramification(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
y = R.laurent_series_ring().gen()
p = x
self.assertEqual(p.laurent_part, y)
self.assertEqual(p.ramification_index, 1)
p = x**2
self.assertEqual(p.laurent_part, y**2)
self.assertEqual(p.ramification_index, 1)
p = x**(QQ(1)/2)
self.assertEqual(p.laurent_part, y)
self.assertEqual(p.ramification_index, 2)
p = x**(QQ(2)/3)
self.assertEqual(p.laurent_part, y**2)
self.assertEqual(p.ramification_index, 3)
p = 1 + 42*x**(QQ(1)/2) + 99*x**(QQ(1)/3)
self.assertEqual(p.laurent_part, 1 + 99*y**2 + 42*y**3)
self.assertEqual(p.ramification_index, 6)
def test_add(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
half = QQ(1) / QQ(2)
p = 1
q = t
r = 1 + t
self.assertEqual(p + q, r)
p = 1
q = t**half
r = 1 + t**half
self.assertEqual(p + q, r)
p = 1 + t
q = 1 + t + t**2
r = 2 + 2 * t + t**2
self.assertEqual(p + q, r)
p = 1 + t**(QQ(1) / 2)
q = 1 + t**(QQ(1) / 2) + t
r = 2 + 2 * t**(QQ(1) / 2) + t
self.assertEqual(p + q, r)
def test_repr(self):
R = PuiseuxSeriesRing(QQ, 't')
t = R.gen()
p = R(1)
s = '1'
self.assertEqual(str(p), s)
p = t
s = 't'
self.assertEqual(str(p), s)
p = t**2
s = 't^2'
self.assertEqual(str(p), s)
half = QQ(1)/QQ(2)
p = t**half
s = 't^(1/2)'
self.assertEqual(str(p), s)
p = t**(-half)
s = 't^(-1/2)'
self.assertEqual(str(p), s)
def test_change_ring(self):
R = PuiseuxSeriesRing(QQ, 'x')
S = R.change_ring(QQbar)
self.assertEqual(R.base_ring(), QQ)
self.assertEqual(S.base_ring(), QQbar)
T = R.change_ring(SR)
self.assertEqual(T.base_ring(), SR)
B = QQ['a,b']
U = R.change_ring(B)
self.assertEqual(U.base_ring(), B)
def test_change_ring(self):
R = PuiseuxSeriesRing(QQ, 'x')
x = R.gen()
half = QQ(1) / 2
p = x**(-half) + 1 + x + half * x**(5 * half)
S = PuiseuxSeriesRing(CC, 'x')
q = p.change_ring(CC)
self.assertEqual(q.parent(), S)
T = PuiseuxSeriesRing(SR, 'x')
r = p.change_ring(SR)
self.assertEqual(r.parent(), T)
B = QQ['a,b']
U = PuiseuxSeriesRing(B, 'x')
s = p.change_ring(B)
self.assertEqual(s.parent(), U)
def test_construction_SR(self):
R = PuiseuxSeriesRing(SR, 'x')
x = R.gen()
def test_construction_QQbar(self):
R = PuiseuxSeriesRing(QQbar, 'x')
x = R.gen()
def test_construction_QQbar(self):
R = PuiseuxSeriesRing(QQbar, 'x')
x = R.gen()
def test_construction_SR(self):
R = PuiseuxSeriesRing(SR, 'x')
x = R.gen()