def __init__(self, PP, f, h=None, names=None, genus=None):
x, y, z = PP.gens()
df = f.degree()
F1 = sum([f[i] * x**i * z**(df - i) for i in range(df + 1)])
if h is None:
F = y**2 * z**(df - 2) - F1
else:
dh = h.degree()
deg = max(df, dh + 1)
F0 = sum([h[i] * x**i * z**(dh - i) for i in range(dh + 1)])
F = y**2 * z**(deg - 2) + F0 * y * z**(deg - dh -
1) - F1 * z**(deg - df)
plane_curve.ProjectivePlaneCurve.__init__(self, PP, F)
R = PP.base_ring()
if names is None:
names = ("x", "y")
else:
names = normalize_names(2, names)
self._names = names
P1 = PolynomialRing(R, name=names[0])
P2 = PolynomialRing(P1, name=names[1])
self._PP = PP
self._printing_ring = P2
self._hyperelliptic_polynomials = (f, h)
self._genus = genus
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 local_coordinates_at_infinity(self, prec=20, name='t'):
"""
For the genus `g` hyperelliptic curve `y^2 = f(x)`, return
`(x(t), y(t))` such that `(y(t))^2 = f(x(t))`, where `t = x^g/y` is
the local parameter at infinity
INPUT:
- ``prec`` -- desired precision of the local coordinates
- ``name`` -- generator of the power series ring (default: ``t``)
OUTPUT:
`(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x^g/y`
is the local parameter at infinity
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-5*x^2+1)
sage: x,y = H.local_coordinates_at_infinity(10)
sage: x
t^-2 + 5*t^4 - t^8 - 50*t^10 + O(t^12)
sage: y
t^-5 + 10*t - 2*t^5 - 75*t^7 + 50*t^11 + O(t^12)
::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-x+1)
sage: x,y = H.local_coordinates_at_infinity(10)
sage: x
t^-2 + t^2 - t^4 - t^6 + 3*t^8 + O(t^12)
sage: y
t^-3 + t - t^3 - t^5 + 3*t^7 - 10*t^11 + O(t^12)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
g = self.genus()
pol = self.hyperelliptic_polynomials()[0]
K = LaurentSeriesRing(self.base_ring(), name, default_prec=prec + 2)
t = K.gen()
L = PolynomialRing(K, 'x')
x = L.gen()
i = 0
w = (x**g / t)**2 - pol
wprime = w.derivative(x)
if pol.degree() == 2 * g + 1:
x = t**-2
else:
x = t**-1
for i in range((RR(log(prec + 2) / log(2))).ceil()):
x = x - w(x) / wprime(x)
y = x**g / t
return x + O(t**(prec + 2)), y + O(t**(prec + 2))
def hyperelliptic_polynomials(self, K=None, var='x'):
"""
EXAMPLES::
sage: R.<x> = QQ[]; C = HyperellipticCurve(x^3 + x - 1, x^3/5); C
Hyperelliptic Curve over Rational Field defined by y^2 + 1/5*x^3*y = x^3 + x - 1
sage: C.hyperelliptic_polynomials()
(x^3 + x - 1, 1/5*x^3)
"""
if K is None:
return self._hyperelliptic_polynomials
else:
f, h = self._hyperelliptic_polynomials
P = PolynomialRing(K, var)
return (P(f), P(h))
def cover_polynomial(self, K=None, var="x"):
"""
Return the polynomial defining the cyclic cover.
EXAMPLES::
sage: ZZx.<x> = ZZ[]; CyclicCover(5, x^5 + x + 1).cover_polynomial()
x^5 + x + 1
"""
if K is None:
return self._f
else:
P = PolynomialRing(K, var)
return P(self._f)
def __init__(self,
base_ring,
num_gens,
name_list,
order='negdeglex',
default_prec=10,
sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
TESTS:
By :trac:`14084`, the multi-variate power series ring belongs to the
category of integral domains, if the base ring does::
sage: P = ZZ[['x','y']]
sage: P.category()
Category of integral domains
sage: TestSuite(P).run()
Otherwise, it belongs to the category of commutative rings::
sage: P = Integers(15)[['x','y']]
sage: P.category()
Category of commutative rings
sage: TestSuite(P).run()
"""
order = TermOrder(order, num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError(
"Multivariate Polynomial Rings must have more than 0 variables."
)
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
# Multivariate power series rings inherit from power series rings. But
# apparently we can not call their initialisation. Instead, initialise
# CommutativeRing and Nonexact:
CommutativeRing.__init__(self,
base_ring,
name_list,
category=_IntegralDomains if base_ring
in _IntegralDomains else _CommutativeRings)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring,
self.variable_names(),
sparse=sparse,
order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_,
'Tbg',
sparse=sparse,
default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list, order, default_prec,
sparse)
self._populate_coercion_lists_()
def splitting_field(poly,
name,
map=False,
degree_multiple=None,
abort_degree=None,
simplify=True,
simplify_all=False):
r"""
Compute the splitting field of a given polynomial, defined over a
number field.
INPUT:
- ``poly`` -- a monic polynomial over a number field
- ``name`` -- a variable name for the number field
- ``map`` -- (default: ``False``) also return an embedding of
``poly`` into the resulting field. Note that computing this
embedding might be expensive.
- ``degree_multiple`` -- a multiple of the absolute degree of
the splitting field. If ``degree_multiple`` equals the actual
degree, this can enormously speed up the computation.
- ``abort_degree`` -- abort by raising a :class:`SplittingFieldAbort`
if it can be determined that the absolute degree of the splitting
field is strictly larger than ``abort_degree``.
- ``simplify`` -- (default: ``True``) during the algorithm, try
to find a simpler defining polynomial for the intermediate
number fields using PARI's ``polred()``. This usually speeds
up the computation but can also considerably slow it down.
Try and see what works best in the given situation.
- ``simplify_all`` -- (default: ``False``) If ``True``, simplify
intermediate fields and also the resulting number field.
OUTPUT:
If ``map`` is ``False``, the splitting field as an absolute number
field. If ``map`` is ``True``, a tuple ``(K, phi)`` where ``phi``
is an embedding of the base field in ``K``.
EXAMPLES::
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = (x^3 + 2).splitting_field(); K
Number Field in a with defining polynomial x^6 + 3*x^5 + 6*x^4 + 11*x^3 + 12*x^2 - 3*x + 1
sage: K.<a> = (x^3 - 3*x + 1).splitting_field(); K
Number Field in a with defining polynomial x^3 - 3*x + 1
The ``simplify`` and ``simplify_all`` flags usually yield
fields defined by polynomials with smaller coefficients.
By default, ``simplify`` is True and ``simplify_all`` is False.
::
sage: (x^4 - x + 1).splitting_field('a', simplify=False)
Number Field in a with defining polynomial x^24 - 2780*x^22 + 2*x^21 + 3527512*x^20 - 2876*x^19 - 2701391985*x^18 + 945948*x^17 + 1390511639677*x^16 + 736757420*x^15 - 506816498313560*x^14 - 822702898220*x^13 + 134120588299548463*x^12 + 362240696528256*x^11 - 25964582366880639486*x^10 - 91743672243419990*x^9 + 3649429473447308439427*x^8 + 14310332927134072336*x^7 - 363192569823568746892571*x^6 - 1353403793640477725898*x^5 + 24293393281774560140427565*x^4 + 70673814899934142357628*x^3 - 980621447508959243128437933*x^2 - 1539841440617805445432660*x + 18065914012013502602456565991
sage: (x^4 - x + 1).splitting_field('a', simplify=True)
Number Field in a with defining polynomial x^24 + 8*x^23 - 32*x^22 - 310*x^21 + 540*x^20 + 4688*x^19 - 6813*x^18 - 32380*x^17 + 49525*x^16 + 102460*x^15 - 129944*x^14 - 287884*x^13 + 372727*x^12 + 150624*x^11 - 110530*x^10 - 566926*x^9 + 1062759*x^8 - 779940*x^7 + 863493*x^6 - 1623578*x^5 + 1759513*x^4 - 955624*x^3 + 459975*x^2 - 141948*x + 53919
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x + 1
Reducible polynomials also work::
sage: pol = (x^4 - 1)*(x^2 + 1/2)*(x^2 + 1/3)
sage: pol.splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^8 - x^4 + 1
Relative situation::
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 + 2)
sage: S.<t> = PolynomialRing(K)
sage: L.<b> = (t^2 - a).splitting_field()
sage: L
Number Field in b with defining polynomial t^6 + 2
With ``map=True``, we also get the embedding of the base field
into the splitting field::
sage: L.<b>, phi = (t^2 - a).splitting_field(map=True)
sage: phi
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To: Number Field in b with defining polynomial t^6 + 2
Defn: a |--> b^2
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True, map=True)[1]
Ring morphism:
From: Rational Field
To: Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x + 1
Defn: 1 |--> 1
We can enable verbose messages::
sage: from sage.misc.verbose import set_verbose
sage: set_verbose(2)
sage: K.<a> = (x^3 - x + 1).splitting_field()
verbose 1 (...: splitting_field.py, splitting_field) Starting field: y
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor: [(3, 0)]
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle: [(2, 2), (3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree: [6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^2 + 23
verbose 1 (...: splitting_field.py, splitting_field) New field before simplifying: x^2 + 23 (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) New field: y^2 - y + 6 (time = ...)
verbose 2 (...: splitting_field.py, splitting_field) Converted polynomials to new field (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor: []
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle: [(3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree: [6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^3 - x + 1
verbose 1 (...: splitting_field.py, splitting_field) New field: y^6 + 3*y^5 + 19*y^4 + 35*y^3 + 127*y^2 + 73*y + 271 (time = ...)
sage: set_verbose(0)
Try all Galois groups in degree 4. We use a quadratic base field
such that ``polgalois()`` cannot be used::
sage: R.<x> = PolynomialRing(QuadraticField(-11))
sage: C2C2pol = x^4 - 10*x^2 + 1
sage: C2C2pol.splitting_field('x')
Number Field in x with defining polynomial x^8 + 24*x^6 + 608*x^4 + 9792*x^2 + 53824
sage: C4pol = x^4 + x^3 + x^2 + x + 1
sage: C4pol.splitting_field('x')
Number Field in x with defining polynomial x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^3 - 18*x^2 - 27*x + 81
sage: D8pol = x^4 - 2
sage: D8pol.splitting_field('x')
Number Field in x with defining polynomial x^16 + 8*x^15 + 68*x^14 + 336*x^13 + 1514*x^12 + 5080*x^11 + 14912*x^10 + 35048*x^9 + 64959*x^8 + 93416*x^7 + 88216*x^6 + 41608*x^5 - 25586*x^4 - 60048*x^3 - 16628*x^2 + 12008*x + 34961
sage: A4pol = x^4 - 4*x^3 + 14*x^2 - 28*x + 21
sage: A4pol.splitting_field('x')
Number Field in x with defining polynomial x^24 - 20*x^23 + 290*x^22 - 3048*x^21 + 26147*x^20 - 186132*x^19 + 1130626*x^18 - 5913784*x^17 + 26899345*x^16 - 106792132*x^15 + 371066538*x^14 - 1127792656*x^13 + 2991524876*x^12 - 6888328132*x^11 + 13655960064*x^10 - 23000783036*x^9 + 32244796382*x^8 - 36347834476*x^7 + 30850889884*x^6 - 16707053128*x^5 + 1896946429*x^4 + 4832907884*x^3 - 3038258802*x^2 - 200383596*x + 593179173
sage: S4pol = x^4 + x + 1
sage: S4pol.splitting_field('x')
Number Field in x with defining polynomial x^48 ...
Some bigger examples::
sage: R.<x> = PolynomialRing(QQ)
sage: pol15 = chebyshev_T(31, x) - 1 # 2^30*(x-1)*minpoly(cos(2*pi/31))^2
sage: pol15.splitting_field('a')
Number Field in a with defining polynomial x^15 - x^14 - 14*x^13 + 13*x^12 + 78*x^11 - 66*x^10 - 220*x^9 + 165*x^8 + 330*x^7 - 210*x^6 - 252*x^5 + 126*x^4 + 84*x^3 - 28*x^2 - 8*x + 1
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
sage: pol48.splitting_field('a')
Number Field in a with defining polynomial x^48 ...
If you somehow know the degree of the field in advance, you
should add a ``degree_multiple`` argument. This can speed up the
computation, in particular for polynomials of degree >= 12 or
for relative extensions::
sage: pol15.splitting_field('a', degree_multiple=15)
Number Field in a with defining polynomial x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 1
A value for ``degree_multiple`` which isn't actually a
multiple of the absolute degree of the splitting field can
either result in a wrong answer or the following exception::
sage: pol48.splitting_field('a', degree_multiple=20)
Traceback (most recent call last):
...
ValueError: inconsistent degree_multiple in splitting_field()
Compute the Galois closure as the splitting field of the defining polynomial::
sage: R.<x> = PolynomialRing(QQ)
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
sage: K.<a> = NumberField(pol48)
sage: L.<b> = pol48.change_ring(K).splitting_field()
sage: L
Number Field in b with defining polynomial x^48 ...
Try all Galois groups over `\QQ` in degree 5 except for `S_5`
(the latter is infeasible with the current implementation)::
sage: C5pol = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: C5pol.splitting_field('x')
Number Field in x with defining polynomial x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: D10pol = x^5 - x^4 - 5*x^3 + 4*x^2 + 3*x - 1
sage: D10pol.splitting_field('x')
Number Field in x with defining polynomial x^10 - 28*x^8 + 216*x^6 - 681*x^4 + 902*x^2 - 401
sage: AGL_1_5pol = x^5 - 2
sage: AGL_1_5pol.splitting_field('x')
Number Field in x with defining polynomial x^20 + 10*x^19 + 55*x^18 + 210*x^17 + 595*x^16 + 1300*x^15 + 2250*x^14 + 3130*x^13 + 3585*x^12 + 3500*x^11 + 2965*x^10 + 2250*x^9 + 1625*x^8 + 1150*x^7 + 750*x^6 + 400*x^5 + 275*x^4 + 100*x^3 + 75*x^2 + 25
sage: A5pol = x^5 - x^4 + 2*x^2 - 2*x + 2
sage: A5pol.splitting_field('x')
Number Field in x with defining polynomial x^60 ...
We can use the ``abort_degree`` option if we don't want to compute
fields of too large degree (this can be used to check whether the
splitting field has small degree)::
sage: (x^5+x+3).splitting_field('b', abort_degree=119)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field equals 120
sage: (x^10+x+3).splitting_field('b', abort_degree=60) # long time (10s on sage.math, 2014)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field is a multiple of 180
Use the ``degree_divisor`` attribute to recover the divisor of the
degree of the splitting field or ``degree_multiple`` to recover a
multiple::
sage: from sage.rings.number_field.splitting_field import SplittingFieldAbort
sage: try: # long time (4s on sage.math, 2014)
....: (x^8+x+1).splitting_field('b', abort_degree=60, simplify=False)
....: except SplittingFieldAbort as e:
....: print(e.degree_divisor)
....: print(e.degree_multiple)
120
1440
TESTS::
sage: from sage.rings.number_field.splitting_field import splitting_field
sage: splitting_field(polygen(QQ), name='x', map=True, simplify_all=True)
(Number Field in x with defining polynomial x, Ring morphism:
From: Rational Field
To: Number Field in x with defining polynomial x
Defn: 1 |--> 1)
"""
from sage.misc.all import cputime
from sage.misc.verbose import verbose
degree_multiple = Integer(degree_multiple or 0)
abort_degree = Integer(abort_degree or 0)
# Kpol = PARI polynomial in y defining the extension found so far
F = poly.base_ring()
if is_RationalField(F):
Kpol = pari("'y")
else:
Kpol = F.pari_polynomial("y")
# Fgen = the generator of F as element of Q[y]/Kpol
# (only needed if map=True)
if map:
Fgen = F.gen().__pari__()
verbose("Starting field: %s" % Kpol)
# L and Lred are lists of SplittingData.
# L contains polynomials which are irreducible over K,
# Lred contains polynomials which need to be factored.
L = []
Lred = [SplittingData(poly._pari_with_name(), degree_multiple)]
# Main loop, handle polynomials one by one
while True:
# Absolute degree of current field K
absolute_degree = Integer(Kpol.poldegree())
# Compute minimum relative degree of splitting field
rel_degree_divisor = Integer(1)
for splitting in L:
rel_degree_divisor = rel_degree_divisor.lcm(splitting.poldegree())
# Check for early aborts
abort_rel_degree = abort_degree // absolute_degree
if abort_rel_degree and rel_degree_divisor > abort_rel_degree:
raise SplittingFieldAbort(absolute_degree * rel_degree_divisor,
degree_multiple)
# First, factor polynomials in Lred and store the result in L
verbose("SplittingData to factor: %s" %
[s._repr_tuple() for s in Lred])
t = cputime()
for splitting in Lred:
m = splitting.dm.gcd(degree_multiple).gcd(
factorial(splitting.poldegree()))
if m == 1:
continue
factors = Kpol.nffactor(splitting.pol)[0]
for q in factors:
d = q.poldegree()
fac = factorial(d)
# Multiple of the degree of the splitting field of q,
# note that the degree equals fac iff the Galois group is S_n.
mq = m.gcd(fac)
if mq == 1:
continue
# Multiple of the degree of the splitting field of q
# over the field defined by adding square root of the
# discriminant.
# If the Galois group is contained in A_n, then mq_alt is
# also the degree multiple over the current field K.
# Here, we have equality if the Galois group is A_n.
mq_alt = mq.gcd(fac // 2)
# If we are over Q, then use PARI's polgalois() to compute
# these degrees exactly.
if absolute_degree == 1:
try:
G = q.polgalois()
except PariError:
pass
else:
mq = Integer(G[0])
mq_alt = mq // 2 if (G[1] == -1) else mq
# In degree 4, use the cubic resolvent to refine the
# degree bounds.
if d == 4 and mq >= 12: # mq equals 12 or 24
# Compute cubic resolvent
a0, a1, a2, a3, a4 = (q / q.pollead()).Vecrev()
assert a4 == 1
cubicpol = pari([
4 * a0 * a2 - a1 * a1 - a0 * a3 * a3, a1 * a3 - 4 * a0,
-a2, 1
]).Polrev()
cubicfactors = Kpol.nffactor(cubicpol)[0]
if len(cubicfactors) == 1: # A4 or S4
# After adding a root of the cubic resolvent,
# the degree of the extension defined by q
# is a factor 3 smaller.
L.append(SplittingData(cubicpol, 3))
rel_degree_divisor = rel_degree_divisor.lcm(3)
mq = mq // 3 # 4 or 8
mq_alt = 4
elif len(cubicfactors) == 2: # C4 or D8
# The irreducible degree 2 factor is
# equivalent to x^2 - q.poldisc().
discpol = cubicfactors[1]
L.append(SplittingData(discpol, 2))
mq = mq_alt = 4
else: # C2 x C2
mq = mq_alt = 4
if mq > mq_alt >= 3:
# Add quadratic resolvent x^2 - D to decrease
# the degree multiple by a factor 2.
discpol = pari([-q.poldisc(), 0, 1]).Polrev()
discfactors = Kpol.nffactor(discpol)[0]
if len(discfactors) == 1:
# Discriminant is not a square
L.append(SplittingData(discpol, 2))
rel_degree_divisor = rel_degree_divisor.lcm(2)
mq = mq_alt
L.append(SplittingData(q, mq))
rel_degree_divisor = rel_degree_divisor.lcm(q.poldegree())
if abort_rel_degree and rel_degree_divisor > abort_rel_degree:
raise SplittingFieldAbort(
absolute_degree * rel_degree_divisor, degree_multiple)
verbose("Done factoring", t, level=2)
if len(L) == 0: # Nothing left to do
break
# Recompute absolute degree multiple
new_degree_multiple = absolute_degree
for splitting in L:
new_degree_multiple *= splitting.dm
degree_multiple = new_degree_multiple.gcd(degree_multiple)
# Absolute degree divisor
degree_divisor = rel_degree_divisor * absolute_degree
# Sort according to degree to handle low degrees first
L.sort(key=lambda x: x.key())
verbose("SplittingData to handle: %s" % [s._repr_tuple() for s in L])
verbose("Bounds for absolute degree: [%s, %s]" %
(degree_divisor, degree_multiple))
# Check consistency
if degree_multiple % degree_divisor != 0:
raise ValueError(
"inconsistent degree_multiple in splitting_field()")
for splitting in L:
# The degree of the splitting field must be a multiple of
# the degree of the polynomial. Only do this check for
# SplittingData with minimal dm, because the higher dm are
# defined as relative degree over the splitting field of
# the polynomials with lesser dm.
if splitting.dm > L[0].dm:
break
if splitting.dm % splitting.poldegree() != 0:
raise ValueError(
"inconsistent degree_multiple in splitting_field()")
# Add a root of f = L[0] to construct the field N = K[x]/f(x)
splitting = L[0]
f = splitting.pol
verbose("Handling polynomial %s" % (f.lift()), level=2)
t = cputime()
Npol, KtoN, k = Kpol.rnfequation(f, flag=1)
# Make Npol monic integral primitive, store in Mpol
# (after this, we don't need Npol anymore, only Mpol)
Mdiv = pari(1)
Mpol = Npol
while True:
denom = Integer(Mpol.pollead())
if denom == 1:
break
denom = pari(denom.factor().radical_value())
Mpol = (Mpol * (denom**Mpol.poldegree())).subst(
"x",
pari([0, 1 / denom]).Polrev("x"))
Mpol /= Mpol.content()
Mdiv *= denom
# We are finished for sure if we hit the degree bound
finished = (Mpol.poldegree() >= degree_multiple)
if simplify_all or (simplify and not finished):
# Find a simpler defining polynomial Lpol for Mpol
verbose("New field before simplifying: %s" % Mpol, t)
t = cputime()
M = Mpol.polred(flag=3)
n = len(M[0]) - 1
Lpol = M[1][n].change_variable_name("y")
LtoM = M[0][n].change_variable_name("y").Mod(
Mpol.change_variable_name("y"))
MtoL = LtoM.modreverse()
else:
# Lpol = Mpol
Lpol = Mpol.change_variable_name("y")
MtoL = pari("'y")
NtoL = MtoL / Mdiv
KtoL = KtoN.lift().subst("x", NtoL).Mod(Lpol)
Kpol = Lpol # New Kpol (for next iteration)
verbose("New field: %s" % Kpol, t)
if map:
t = cputime()
Fgen = Fgen.lift().subst("y", KtoL)
verbose("Computed generator of F in K", t, level=2)
if finished:
break
t = cputime()
# Convert f and elements of L from K to L and store in L
# (if the polynomial is certain to remain irreducible) or Lred.
Lold = L[1:]
L = []
Lred = []
# First add f divided by the linear factor we obtained,
# mg is the new degree multiple.
mg = splitting.dm // f.poldegree()
if mg > 1:
g = [c.subst("y", KtoL).Mod(Lpol) for c in f.Vecrev().lift()]
g = pari(g).Polrev()
g /= pari([k * KtoL - NtoL, 1]).Polrev() # divide linear factor
Lred.append(SplittingData(g, mg))
for splitting in Lold:
g = [c.subst("y", KtoL) for c in splitting.pol.Vecrev().lift()]
g = pari(g).Polrev()
mg = splitting.dm
if Integer(g.poldegree()).gcd(
f.poldegree()) == 1: # linearly disjoint fields
L.append(SplittingData(g, mg))
else:
Lred.append(SplittingData(g, mg))
verbose("Converted polynomials to new field", t, level=2)
# Convert Kpol to Sage and construct the absolute number field
Kpol = PolynomialRing(RationalField(),
name=poly.variable_name())(Kpol / Kpol.pollead())
K = NumberField(Kpol, name)
if map:
return K, F.hom(Fgen, K)
else:
return K
def mnuk_conditions(g, b, generic_adjoint):
"""Determine the Mnuk conditions on the coefficients of :math:`P`.
Determine the conditions on the coefficients `c` of `P` at the integral
basis element `b` modulo the curve `g = g(u,v)`. See [Mnuk] for details.
Parameters
----------
g : curve
An algebraic curve.
b : integral basis function
An an element of the basis of the integral closure of the coordinate
ring of `g`. See :func:`abelfunctions.integralbasis.integral_basis`.
generic_adjoint : polynomial
A generic adjoint polynomial as provided by :func:`differentials`. Only
one instance is created for caching and performance purposes.
Returns
-------
conditions : list
A list of expressions from which a system of equations is build to
determine the differentials.
"""
# extract rings. the generic adjoint should be a member of R[*c][u,v] where
# *c is a vector of the indeterminants. we will need to convert it to a
# polynomial in R[u,v,*c] and then back (see below)
R = g.parent()
S = generic_adjoint.parent()
B = S.base_ring()
c = B.gens()
T = QQbar[R.variable_names() + B.variable_names()]
# compute b_num(x,y) * P(x,y) and reduce modulo the defining polynomial g.
# we do this by casting the polynomial into the ring QQbar(x,*c)[y]. (the
# coefficients of y in g need to be units)
B = PolynomialRing(QQbar, [R.variable_names()[0]] + list(B.variable_names()))
Q = B.fraction_field()[R.variable_names()[1]]
u,v = map(Q,R.gens())
numer = b.numerator()
denom = b.denominator()
expr = numer(u,v) * generic_adjoint(u,v)
modulus = g(u,v)
r_reduced_mod_g = expr % modulus
# now mod out by the denominator to get the remaining component, R(x,y). we
# need to cast into the ring QQbar[y,*c][x] in order to do so. (note that
# we don't need a base fraction field since the denominator is univariate
# and therefore the leading coefficient is always a unit)
u,v = map(T, R.gens())
r = r_reduced_mod_g(v).numerator()
r_reduced_mod_denom = r.polynomial(u) % T(denom).polynomial(u)
# finally, coerce the result to QQbar[*c][x,y] in order to obtain the
# coefficients as linear combinations of the c_ij's.
r = T(r_reduced_mod_denom(u)) # first need to coerce to "largest" ring, T
u,v = map(S, R.gens())
c = map(S, c)
args = [u, v] + c
r = r(*args)
conditions = r.coefficients()
return conditions
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 rational_points_iterator(self):
r"""
Return a generator object for the rational points on this curve.
INPUT:
- ``self`` -- a projective curve
OUTPUT:
A generator of all the rational points on the curve defined over its base field.
EXAMPLE::
sage: F = GF(37)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^7+Y*X*Z^5*55+Y^7*12)
sage: len(list(C.rational_points_iterator()))
37
::
sage: F = GF(2)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X*Y*Z)
sage: a = C.rational_points_iterator()
sage: a.next()
(1 : 0 : 0)
sage: a.next()
(0 : 1 : 0)
sage: a.next()
(1 : 1 : 0)
sage: a.next()
(0 : 0 : 1)
sage: a.next()
(1 : 0 : 1)
sage: a.next()
(0 : 1 : 1)
sage: a.next()
Traceback (most recent call last):
...
StopIteration
::
sage: F = GF(3^2,'a')
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^3+5*Y^2*Z-33*X*Y*X)
sage: b = C.rational_points_iterator()
sage: b.next()
(0 : 1 : 0)
sage: b.next()
(0 : 0 : 1)
sage: b.next()
(2*a + 2 : a : 1)
sage: b.next()
(2 : a + 1 : 1)
sage: b.next()
(a + 1 : 2*a + 1 : 1)
sage: b.next()
(1 : 2 : 1)
sage: b.next()
(2*a + 2 : 2*a : 1)
sage: b.next()
(2 : 2*a + 2 : 1)
sage: b.next()
(a + 1 : a + 2 : 1)
sage: b.next()
(1 : 1 : 1)
sage: b.next()
Traceback (most recent call last):
...
StopIteration
"""
g = self.defining_polynomial()
K = g.parent().base_ring()
from sage.rings.polynomial.all import PolynomialRing
R = PolynomialRing(K, 'X')
X = R.gen()
one = K.one_element()
zero = K.zero_element()
# the point with Z = 0 = Y
try:
t = self.point([one, zero, zero])
yield (t)
except TypeError:
pass
# points with Z = 0, Y = 1
g10 = R(g(X, one, zero))
if g10.is_zero():
for x in K:
yield (self.point([x, one, zero]))
else:
for x in g10.roots(multiplicities=False):
yield (self.point([x, one, zero]))
# points with Z = 1
for y in K:
gy1 = R(g(X, y, one))
if gy1.is_zero():
for x in K:
yield (self.point([x, y, one]))
else:
for x in gy1.roots(multiplicities=False):
yield (self.point([x, y, one]))
def local_coordinates(self, pt, n):
r"""
Return local coordinates to precision n at the given point.
Behaviour is flaky - some choices of `n` are worst that
others.
INPUT:
- ``pt`` - an F-rational point on X which is not a
point of ramification for the projection (x,y) - x.
- ``n`` - the number of terms desired
OUTPUT: x = x0 + t y = y0 + power series in t
EXAMPLES::
sage: FF = FiniteField(5)
sage: P2 = ProjectiveSpace(2, FF, names = ['x','y','z'])
sage: x, y, z = P2.coordinate_ring().gens()
sage: C = Curve(y^2*z^7-x^9-x*z^8)
sage: pt = C([2,3,1])
sage: C.local_coordinates(pt,9) # todo: not implemented !!!!
[2 + t, 3 + 3*t^2 + t^3 + 3*t^4 + 3*t^6 + 3*t^7 + t^8 + 2*t^9 + 3*t^11 + 3*t^12]
"""
f = self.defining_polynomial()
R = f.parent()
F = self.base_ring()
p = F.characteristic()
x0 = F(pt[0])
y0 = F(pt[1])
astr = ["a" + str(i) for i in range(1, 2 * n)]
x, y = R.gens()
R0 = PolynomialRing(F, 2 * n + 2, names=[str(x), str(y), "t"] + astr)
vars0 = R0.gens()
t = vars0[2]
yt = y0 * t**0 + add(
[vars0[i] * t**(i - 2) for i in range(3, 2 * n + 2)])
xt = x0 + t
ft = f(xt, yt)
S = singular
S.eval('ring s = ' + str(p) + ',' + str(R0.gens()) + ',lp;')
S.eval('poly f = ' + str(ft))
cmd = 'matrix c = coeffs (' + str(ft) + ',t)'
S.eval(cmd)
N = int(S.eval('size(c)'))
b = ["c[" + str(i) + ",1]," for i in range(2, N / 2 - 4)]
b = ''.join(b)
b = b[:len(b) - 1] #to cut off the trailing comma
cmd = 'ideal I = ' + b
S.eval(cmd)
c = S.eval('slimgb(I)')
d = c.split("=")
d = d[1:]
d[len(d) - 1] += "\n"
e = [x[:x.index("\n")] for x in d]
vals = []
for x in e:
for y in vars0:
if str(y) in x:
if len(x.replace(str(y), "")) != 0:
i = x.find("-")
if i > 0:
vals.append(
[eval(x[1:i]), x[:i],
F(eval(x[i + 1:]))])
i = x.find("+")
if i > 0:
vals.append(
[eval(x[1:i]), x[:i], -F(eval(x[i + 1:]))])
else:
vals.append([eval(str(y)[1:]), str(y), F(0)])
vals.sort()
k = len(vals)
v = [x0 + t, y0 + add([vals[i][2] * t**(i + 1) for i in range(k)])]
return v
def __init__(self,
base_ring,
num_gens,
name_list,
order='negdeglex',
default_prec=10,
sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
"""
order = TermOrder(order, num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError(
"Multivariate Polynomial Rings must have more than 0 variables."
)
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
ParentWithGens.__init__(self, base_ring, name_list)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring,
self.variable_names(),
sparse=sparse,
order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_,
'Tbg',
sparse=sparse,
default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
## use the following in PowerSeriesRing_generic.__call__
self._PowerSeriesRing_generic__power_series_class = MPowerSeries
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list, order, default_prec,
sparse)
self._populate_coercion_lists_()