def __init__(self, W, q, trace=False):
"""
A class for Kazhdan-Lusztig polynomials
INPUT:
- ``W`` - a Weyl Group.
- ``q`` - an indeterminate.
OPTIONAL:
- trace
The parent of q may be a PolynomialRing or a LaurentPolynomialRing.
Set trace=True in order to see intermediate results. This is fun
and instructive.
EXAMPLES ::
sage: W = WeylGroup("B3",prefix="s")
sage: [s1,s2,s3]=W.simple_reflections()
sage: R.<q>=LaurentPolynomialRing(QQ)
sage: KL=KazhdanLusztigPolynomial(W,q)
sage: KL.P(s2,s3*s2*s3*s1*s2)
q + 1
A faster implementation (using the optional package Coxeter 3) is given by::
sage: W = CoxeterGroup(['B', 3], implementation='coxeter3') # optional - coxeter3
sage: W.kazhdan_lusztig_polynomial([2], [3,2,3,1,2]) # optional - coxeter3
q + 1
"""
self._coxeter_group = W
self._q = q
self._trace = trace
self._one = W.one()
self._base_ring = q.parent()
if is_Polynomial(q):
self._base_ring_type = "polynomial"
elif isinstance(q, LaurentPolynomial_mpair):
self._base_ring_type = "laurent"
else:
self._base_ring_type = "unknown"
def __init__(self, W, q, trace=False):
"""
A class for Kazhdan-Lusztig polynomials
INPUT:
- ``W`` - a Weyl Group.
- ``q`` - an indeterminate.
OPTIONAL:
- trace
The parent of q may be a PolynomialRing or a LaurentPolynomialRing.
Set trace=True in order to see intermediate results. This is fun
and instructive.
EXAMPLES ::
sage: W = WeylGroup("B3",prefix="s")
sage: [s1,s2,s3]=W.simple_reflections()
sage: R.<q>=LaurentPolynomialRing(QQ)
sage: KL=KazhdanLusztigPolynomial(W,q)
sage: KL.P(s2,s3*s2*s3*s1*s2)
q + 1
A faster implementation (using the optional package Coxeter 3) is given by::
sage: W = CoxeterGroup(['B', 3], implementation='coxeter3') # optional - coxeter3
sage: W.kazhdan_lusztig_polynomial([2], [3,2,3,1,2]) # optional - coxeter3
q + 1
"""
self._coxeter_group = W
self._q = q
self._trace = trace
self._one = W.one()
self._base_ring = q.parent()
if is_Polynomial(q):
self._base_ring_type = "polynomial"
elif isinstance(q, LaurentPolynomial_mpair):
self._base_ring_type = "laurent"
else:
self._base_ring_type = "unknown"
def HyperellipticCurve(f,h=None,names=None,PP=None):
r"""
Returns the hyperelliptic curve `y^2 + h y = f`, for
univariate polynomials `h` and `f`. If `h`
is not given, then it defaults to 0.
INPUT:
- ``f`` - univariate polynomial
- ``h`` - optional univariate polynomial
EXAMPLES: A curve with and without the h term::
sage: R.<x> = QQ[]
sage: HyperellipticCurve(x^5 + x + 1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: HyperellipticCurve(x^19 + x + 1, x-2)
Hyperelliptic Curve over Rational Field defined by y^2 + (x - 2)*y = x^19 + x + 1
A curve over a non-prime finite field::
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a)
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by y^2 + (x + a)*y = x^3 + x + 2
Here's one where we change the names of the vars in the homogeneous
polynomial::
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a, names=['X','Y'])
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by Y^2 + (X + a)*Y = X^3 + X + 2
"""
if not is_Polynomial(f):
raise TypeError, "Arguments f (=%s) and h (= %s) must be polynomials"%(f,h)
P = f.parent()
if h is None:
h = P(0)
g = (f.degree()-1)%2
try:
h = P(h)
except TypeError:
raise TypeError, \
"Arguments f (=%s) and h (= %s) must be polynomials in the same ring"%(f,h)
df = f.degree()
dh_2 = 2*h.degree()
if dh_2 < df:
g = (df-1)//2
elif df < dh_2:
g = (dh_2-1)//2
else:
a0 = f.leading_coefficient()
b0 = h.leading_coefficient()
A0 = 4*a0 + b0^2
if A0 != 0:
g = (df-1)//2
else:
if P(2) == 0:
raise TypeError, "Arguments define a curve with finite singularity."
f0 = 4*f + h^2
d0 = f0.degree()
g = (d0-1)//2
R = P.base_ring()
PP = ProjectiveSpace(2, R)
if names is None:
names = ["x","y"]
if is_FiniteField(R):
if g == 2:
return HyperellipticCurve_g2_finite_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_finite_field(PP, f, h, names=names, genus=g)
elif is_RationalField(R):
if g == 2:
return HyperellipticCurve_g2_rational_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_rational_field(PP, f, h, names=names, genus=g)
elif is_pAdicField(R):
if g == 2:
return HyperellipticCurve_g2_padic_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_padic_field(PP, f, h, names=names, genus=g)
else:
if g == 2:
return HyperellipticCurve_g2_generic(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_generic(PP, f, h, names=names, genus=g)
def HyperellipticCurve(f, h=None, names=None, PP=None, check_squarefree=True):
r"""
Returns the hyperelliptic curve `y^2 + h y = f`, for
univariate polynomials `h` and `f`. If `h`
is not given, then it defaults to 0.
INPUT:
- ``f`` - univariate polynomial
- ``h`` - optional univariate polynomial
- ``names`` (default: ``["x","y"]``) - names for the
coordinate functions
- ``check_squarefree`` (default: ``True``) - test if
the input defines a hyperelliptic curve when f is
homogenized to degree `2g+2` and h to degree
`g+1` for some g.
.. WARNING::
When setting ``check_squarefree=False`` or using a base ring that is
not a field, the output curves are not to be trusted. For example, the
output of ``is_singular`` is always ``False``, without this being
properly tested in that case.
.. NOTE::
The words "hyperelliptic curve" are normally only used for curves of
genus at least two, but this class allows more general smooth double
covers of the projective line (conics and elliptic curves), even though
the class is not meant for those and some outputs may be incorrect.
EXAMPLES:
Basic examples::
sage: R.<x> = QQ[]
sage: HyperellipticCurve(x^5 + x + 1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: HyperellipticCurve(x^19 + x + 1, x-2)
Hyperelliptic Curve over Rational Field defined by y^2 + (x - 2)*y = x^19 + x + 1
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a)
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by y^2 + (x + a)*y = x^3 + x + 2
Characteristic two::
sage: P.<x> = GF(8,'a')[]
sage: HyperellipticCurve(x^7+1, x)
Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + x*y = x^7 + 1
sage: HyperellipticCurve(x^8+x^7+1, x^4+1)
Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + (x^4 + 1)*y = x^8 + x^7 + 1
sage: HyperellipticCurve(x^8+1, x)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(x^8+x^7+1, x^4)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: F.<t> = PowerSeriesRing(FiniteField(2))
sage: P.<x> = PolynomialRing(FractionField(F))
sage: HyperellipticCurve(x^5+t, x)
Hyperelliptic Curve over Laurent Series Ring in t over Finite Field of size 2 defined by y^2 + x*y = x^5 + t
We can change the names of the variables in the output::
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a, names=['X','Y'])
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by Y^2 + (X + a)*Y = X^3 + X + 2
This class also allows curves of genus zero or one, which are strictly
speaking not hyperelliptic::
sage: P.<x> = QQ[]
sage: HyperellipticCurve(x^2+1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^2 + 1
sage: HyperellipticCurve(x^4-1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^4 - 1
sage: HyperellipticCurve(x^3+2*x+2)
Hyperelliptic Curve over Rational Field defined by y^2 = x^3 + 2*x + 2
Double roots::
sage: P.<x> = GF(7)[]
sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1))
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1), check_squarefree=False)
Hyperelliptic Curve over Finite Field of size 7 defined by y^2 = x^12 + 5*x^10 + 4*x^9 + x^8 + 3*x^7 + 3*x^6 + 2*x^4 + 3*x^3 + 6*x^2 + 4*x + 3
The input for a (smooth) hyperelliptic curve of genus `g` should not
contain polynomials of degree greater than `2g+2`. In the following
example, the hyperelliptic curve has genus 2 and there exists a model
`y^2 = F` of degree 6, so the model `y^2 + yh = f` of degree 200 is not
allowed.::
sage: P.<x> = QQ[]
sage: h = x^100
sage: F = x^6+1
sage: f = F-h^2/4
sage: HyperellipticCurve(f, h)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(F)
Hyperelliptic Curve over Rational Field defined by y^2 = x^6 + 1
An example with a singularity over an inseparable extension of the
base field::
sage: F.<t> = GF(5)[]
sage: P.<x> = F[]
sage: HyperellipticCurve(x^5+t)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
Input with integer coefficients creates objects with the integers
as base ring, but only checks smoothness over `\QQ`, not over Spec(`\ZZ`).
In other words, it is checked that the discriminant is non-zero, but it is
not checked whether the discriminant is a unit in `\ZZ^*`.::
sage: P.<x> = ZZ[]
sage: HyperellipticCurve(3*x^7+6*x+6)
Hyperelliptic Curve over Integer Ring defined by y^2 = 3*x^7 + 6*x + 6
"""
if (not is_Polynomial(f)) or f == 0:
raise TypeError, "Arguments f (=%s) and h (= %s) must be polynomials " \
"and f must be non-zero" % (f,h)
P = f.parent()
if h is None:
h = P(0)
try:
h = P(h)
except TypeError:
raise TypeError, \
"Arguments f (=%s) and h (= %s) must be polynomials in the same ring"%(f,h)
df = f.degree()
dh_2 = 2*h.degree()
if dh_2 < df:
g = (df-1)//2
else:
g = (dh_2-1)//2
if check_squarefree:
# Assuming we are working over a field, this checks that after
# resolving the singularity at infinity, we get a smooth double cover
# of P^1.
if P(2) == 0:
# characteristic 2
if h == 0:
raise ValueError, \
"In characteristic 2, argument h (= %s) must be non-zero."%h
if h[g+1] == 0 and f[2*g+1]**2 == f[2*g+2]*h[g]**2:
raise ValueError, "Not a hyperelliptic curve: " \
"highly singular at infinity."
should_be_coprime = [h, f*h.derivative()**2+f.derivative()**2]
else:
# characteristic not 2
F = f + h**2/4
if not F.degree() in [2*g+1, 2*g+2]:
raise ValueError, "Not a hyperelliptic curve: " \
"highly singular at infinity."
should_be_coprime = [F, F.derivative()]
try:
smooth = should_be_coprime[0].gcd(should_be_coprime[1]).degree()==0
except (AttributeError, NotImplementedError, TypeError):
try:
smooth = should_be_coprime[0].resultant(should_be_coprime[1])!=0
except (AttributeError, NotImplementedError, TypeError):
raise NotImplementedError, "Cannot determine whether " \
"polynomials %s have a common root. Use " \
"check_squarefree=False to skip this check." % \
should_be_coprime
if not smooth:
raise ValueError, "Not a hyperelliptic curve: " \
"singularity in the provided affine patch."
R = P.base_ring()
PP = ProjectiveSpace(2, R)
if names is None:
names = ["x","y"]
if is_FiniteField(R):
if g == 2:
return HyperellipticCurve_g2_finite_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_finite_field(PP, f, h, names=names, genus=g)
elif is_RationalField(R):
if g == 2:
return HyperellipticCurve_g2_rational_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_rational_field(PP, f, h, names=names, genus=g)
elif is_pAdicField(R):
if g == 2:
return HyperellipticCurve_g2_padic_field(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_padic_field(PP, f, h, names=names, genus=g)
else:
if g == 2:
return HyperellipticCurve_g2_generic(PP, f, h, names=names, genus=g)
else:
return HyperellipticCurve_generic(PP, f, h, names=names, genus=g)
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P, (int, long, Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif isinstance(P, (list, tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1), R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(
self, tuple([P1, P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError, "Argument P (= %s) must have length 2." % P
elif isinstance(
P,
JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]
y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x - x0, R(y0)))
raise TypeError, "Argument P (= %s) does not determine a divisor class" % P
def __call__(self, P):
r"""
Returns a rational point P in the abstract Homset J(K), given:
0. A point P in J = Jac(C), returning P; 1. A point P on the curve
C such that J = Jac(C), where C is an odd degree model, returning
[P - oo]; 2. A pair of points (P, Q) on the curve C such that J =
Jac(C), returning [P-Q]; 2. A list of polynomials (a,b) such that
`b^2 + h*b - f = 0 mod a`, returning [(a(x),y-b(x))].
EXAMPLES::
sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: P = C(0,1,1)
sage: J = C.jacobian()
sage: Q = J(QQ)(P)
sage: for i in range(6): i*Q
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
::
sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)
"""
if isinstance(P,(int,long,Integer)) and P == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif isinstance(P,(list,tuple)):
if len(P) == 1 and P[0] == 0:
R = PolynomialRing(self.value_ring(), 'x')
return JacobianMorphism_divisor_class_field(self, (R(1),R(0)))
elif len(P) == 2:
P1 = P[0]
P2 = P[1]
if is_Integer(P1) and is_Integer(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P1) and is_Polynomial(P2):
R = PolynomialRing(self.value_ring(), 'x')
P1 = R(P1)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Integer(P2) and is_Polynomial(P1):
R = PolynomialRing(self.value_ring(), 'x')
P2 = R(P2)
return JacobianMorphism_divisor_class_field(self, tuple([P1,P2]))
if is_Polynomial(P1) and is_Polynomial(P2):
return JacobianMorphism_divisor_class_field(self, tuple(P))
if is_SchemeMorphism(P1) and is_SchemeMorphism(P2):
return self(P1) - self(P2)
raise TypeError, "Argument P (= %s) must have length 2."%P
elif isinstance(P,JacobianMorphism_divisor_class_field) and self == P.parent():
return P
elif is_SchemeMorphism(P):
x0 = P[0]; y0 = P[1]
R, x = PolynomialRing(self.value_ring(), 'x').objgen()
return self((x-x0,R(y0)))
raise TypeError, "Argument P (= %s) does not determine a divisor class"%P
def HyperellipticCurve(f, h=None, names=None, PP=None, check_squarefree=True):
r"""
Returns the hyperelliptic curve `y^2 + h y = f`, for
univariate polynomials `h` and `f`. If `h`
is not given, then it defaults to 0.
INPUT:
- ``f`` - univariate polynomial
- ``h`` - optional univariate polynomial
- ``names`` (default: ``["x","y"]``) - names for the
coordinate functions
- ``check_squarefree`` (default: ``True``) - test if
the input defines a hyperelliptic curve when f is
homogenized to degree `2g+2` and h to degree
`g+1` for some g.
.. WARNING::
When setting ``check_squarefree=False`` or using a base ring that is
not a field, the output curves are not to be trusted. For example, the
output of ``is_singular`` is always ``False``, without this being
properly tested in that case.
.. NOTE::
The words "hyperelliptic curve" are normally only used for curves of
genus at least two, but this class allows more general smooth double
covers of the projective line (conics and elliptic curves), even though
the class is not meant for those and some outputs may be incorrect.
EXAMPLES:
Basic examples::
sage: R.<x> = QQ[]
sage: HyperellipticCurve(x^5 + x + 1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: HyperellipticCurve(x^19 + x + 1, x-2)
Hyperelliptic Curve over Rational Field defined by y^2 + (x - 2)*y = x^19 + x + 1
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a)
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by y^2 + (x + a)*y = x^3 + x + 2
Characteristic two::
sage: P.<x> = GF(8,'a')[]
sage: HyperellipticCurve(x^7+1, x)
Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + x*y = x^7 + 1
sage: HyperellipticCurve(x^8+x^7+1, x^4+1)
Hyperelliptic Curve over Finite Field in a of size 2^3 defined by y^2 + (x^4 + 1)*y = x^8 + x^7 + 1
sage: HyperellipticCurve(x^8+1, x)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(x^8+x^7+1, x^4)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: F.<t> = PowerSeriesRing(FiniteField(2))
sage: P.<x> = PolynomialRing(FractionField(F))
sage: HyperellipticCurve(x^5+t, x)
Hyperelliptic Curve over Laurent Series Ring in t over Finite Field of size 2 defined by y^2 + x*y = x^5 + t
We can change the names of the variables in the output::
sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a, names=['X','Y'])
Hyperelliptic Curve over Finite Field in a of size 3^2 defined by Y^2 + (X + a)*Y = X^3 + X + 2
This class also allows curves of genus zero or one, which are strictly
speaking not hyperelliptic::
sage: P.<x> = QQ[]
sage: HyperellipticCurve(x^2+1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^2 + 1
sage: HyperellipticCurve(x^4-1)
Hyperelliptic Curve over Rational Field defined by y^2 = x^4 - 1
sage: HyperellipticCurve(x^3+2*x+2)
Hyperelliptic Curve over Rational Field defined by y^2 = x^3 + 2*x + 2
Double roots::
sage: P.<x> = GF(7)[]
sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1))
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
sage: HyperellipticCurve((x^3-x+2)^2*(x^6-1), check_squarefree=False)
Hyperelliptic Curve over Finite Field of size 7 defined by y^2 = x^12 + 5*x^10 + 4*x^9 + x^8 + 3*x^7 + 3*x^6 + 2*x^4 + 3*x^3 + 6*x^2 + 4*x + 3
The input for a (smooth) hyperelliptic curve of genus `g` should not
contain polynomials of degree greater than `2g+2`. In the following
example, the hyperelliptic curve has genus 2 and there exists a model
`y^2 = F` of degree 6, so the model `y^2 + yh = f` of degree 200 is not
allowed.::
sage: P.<x> = QQ[]
sage: h = x^100
sage: F = x^6+1
sage: f = F-h^2/4
sage: HyperellipticCurve(f, h)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: highly singular at infinity.
sage: HyperellipticCurve(F)
Hyperelliptic Curve over Rational Field defined by y^2 = x^6 + 1
An example with a singularity over an inseparable extension of the
base field::
sage: F.<t> = GF(5)[]
sage: P.<x> = F[]
sage: HyperellipticCurve(x^5+t)
Traceback (most recent call last):
...
ValueError: Not a hyperelliptic curve: singularity in the provided affine patch.
Input with integer coefficients creates objects with the integers
as base ring, but only checks smoothness over `\QQ`, not over Spec(`\ZZ`).
In other words, it is checked that the discriminant is non-zero, but it is
not checked whether the discriminant is a unit in `\ZZ^*`.::
sage: P.<x> = ZZ[]
sage: HyperellipticCurve(3*x^7+6*x+6)
Hyperelliptic Curve over Integer Ring defined by y^2 = 3*x^7 + 6*x + 6
"""
if (not is_Polynomial(f)) or f == 0:
raise TypeError, "Arguments f (=%s) and h (= %s) must be polynomials " \
"and f must be non-zero" % (f,h)
P = f.parent()
if h is None:
h = P(0)
try:
h = P(h)
except TypeError:
raise TypeError, \
"Arguments f (=%s) and h (= %s) must be polynomials in the same ring"%(f,h)
df = f.degree()
dh_2 = 2 * h.degree()
if dh_2 < df:
g = (df - 1) // 2
else:
g = (dh_2 - 1) // 2
if check_squarefree:
# Assuming we are working over a field, this checks that after
# resolving the singularity at infinity, we get a smooth double cover
# of P^1.
if P(2) == 0:
# characteristic 2
if h == 0:
raise ValueError, \
"In characteristic 2, argument h (= %s) must be non-zero."%h
if h[g + 1] == 0 and f[2 * g + 1]**2 == f[2 * g + 2] * h[g]**2:
raise ValueError, "Not a hyperelliptic curve: " \
"highly singular at infinity."
should_be_coprime = [h, f * h.derivative()**2 + f.derivative()**2]
else:
# characteristic not 2
F = f + h**2 / 4
if not F.degree() in [2 * g + 1, 2 * g + 2]:
raise ValueError, "Not a hyperelliptic curve: " \
"highly singular at infinity."
should_be_coprime = [F, F.derivative()]
try:
smooth = should_be_coprime[0].gcd(
should_be_coprime[1]).degree() == 0
except (AttributeError, NotImplementedError, TypeError):
try:
smooth = should_be_coprime[0].resultant(
should_be_coprime[1]) != 0
except (AttributeError, NotImplementedError, TypeError):
raise NotImplementedError, "Cannot determine whether " \
"polynomials %s have a common root. Use " \
"check_squarefree=False to skip this check." % \
should_be_coprime
if not smooth:
raise ValueError, "Not a hyperelliptic curve: " \
"singularity in the provided affine patch."
R = P.base_ring()
PP = ProjectiveSpace(2, R)
if names is None:
names = ["x", "y"]
if is_FiniteField(R):
if g == 2:
return HyperellipticCurve_g2_finite_field(PP,
f,
h,
names=names,
genus=g)
else:
return HyperellipticCurve_finite_field(PP,
f,
h,
names=names,
genus=g)
elif is_RationalField(R):
if g == 2:
return HyperellipticCurve_g2_rational_field(PP,
f,
h,
names=names,
genus=g)
else:
return HyperellipticCurve_rational_field(PP,
f,
h,
names=names,
genus=g)
elif is_pAdicField(R):
if g == 2:
return HyperellipticCurve_g2_padic_field(PP,
f,
h,
names=names,
genus=g)
else:
return HyperellipticCurve_padic_field(PP,
f,
h,
names=names,
genus=g)
else:
if g == 2:
return HyperellipticCurve_g2_generic(PP,
f,
h,
names=names,
genus=g)
else:
return HyperellipticCurve_generic(PP, f, h, names=names, genus=g)
