def number_field_to_AA(a):
r"""
It is a mess to convert an element of a number field to the algebraic field
``AA``. This is a temporary fix.
"""
try:
return AA(a)
except TypeError:
return AA.polynomial_root(a.minpoly(), RIF(a))
def from_mathematica(a):
try:
return QQ(a.sage())
except Exception:
pass
try:
return AA(a.sage())
except Exception:
coefficients = mathematica.CoefficientList(
mathematica.MinimalPolynomial(a, 'x'), 'x').sage()
x = polygen(QQ)
minpoly = x.parent()(coefficients)
interval = mathematica.IsolatingInterval(a).sage()
rif_interval = RIF(interval)
return AA.polynomial_root(minpoly, rif_interval)
def flipper_nf_to_sage(K, name='a'):
r"""
Convert a flipper number field into a Sage number field
.. NOTE::
Currently, the code is not careful at all with root isolation.
EXAMPLES::
sage: import flipper # optional - flipper
sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_to_sage
sage: p = flipper.kernel.Polynomial([-2r] + [0r]*5 + [1r]) # optional - flipper
sage: r1,r2 = p.real_roots() # optional - flipper
sage: K = flipper.kernel.NumberField(r1) # optional - flipper
sage: K_sage = flipper_nf_to_sage(K) # optional - flipper
sage: K_sage # optional - flipper
Number Field in a with defining polynomial x^6 - 2
sage: AA(K_sage.gen()) # optional - flipper
-1.122462048309373?
"""
from sage.rings.number_field.number_field import NumberField
from sage.rings.all import QQ,RIF,AA
r = K.lmbda.interval_approximation()
l = r.lower * ZZ(10)**(-r.precision)
u = r.upper * ZZ(10)**(-r.precision)
p = QQ['x'](K.polynomial.coefficients)
s = AA.polynomial_root(p, RIF(l,u))
return NumberField(p, name, embedding=s)
def number_field_elements_from_algebraics(elts, name='a'):
r"""
The native Sage function ``number_field_elements_from_algebraics`` currently
returns number field *without* embedding. This function return field with
embedding!
EXAMPLES::
sage: from flatsurf.geometry.subfield import number_field_elements_from_algebraics
sage: z = QQbar.zeta(5)
sage: c = z.real()
sage: s = z.imag()
sage: number_field_elements_from_algebraics((c,s))
(Number Field in a with defining polynomial y^4 - 5*y^2 + 5 with a = 1.902113032590308?,
[1/2*a^2 - 3/2, 1/2*a])
sage: number_field_elements_from_algebraics([AA(1), AA(2/3)])
(Rational Field, [1, 2/3])
"""
# case when all elements are rationals
if all(x in QQ for x in elts):
return QQ, [QQ(x) for x in elts]
# general case
from sage.rings.qqbar import number_field_elements_from_algebraics
field, elts, phi = number_field_elements_from_algebraics(elts,
minimal=True)
polys = [x.polynomial() for x in elts]
K = NumberField(field.polynomial(), name, embedding=AA(phi(field.gen())))
gen = K.gen()
return K, [x.polynomial()(gen) for x in elts]
def rho(self):
r"""
Return the vertex ``rho`` of the basic hyperbolic
triangle which describes ``self``. ``rho`` has
absolute value 1 and angle ``pi/n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(3).rho() == 1/2 + sqrt(3)/2*i
True
sage: HeckeTriangleGroup(4).rho() == sqrt(2)/2*(1 + i)
True
sage: HeckeTriangleGroup(6).rho() == sqrt(3)/2 + 1/2*i
True
sage: HeckeTriangleGroup(10).rho()
0.95105651629515...? + 0.30901699437494...?*I
sage: HeckeTriangleGroup(infinity).rho()
1
"""
# TODO: maybe rho should be replaced by -rhobar
if (self._n == infinity):
return AA(1)
else:
return AlgebraicField()(exp(pi/self._n*i))
def flipper_nf_to_sage(K, name='a'):
r"""
Convert a flipper number field into a Sage number field
.. NOTE::
Currently, the code is not careful at all with root isolation.
EXAMPLES::
sage: import flipper # optional - flipper
sage: import realalg # optional - flipper
sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_to_sage
sage: K = realalg.RealNumberField([-2r] + [0r]*5 + [1r]) # optional - flipper
sage: K_sage = flipper_nf_to_sage(K) # optional - flipper
sage: K_sage # optional - flipper
Number Field in a with defining polynomial x^6 - 2 with a = 1.122462048309373?
sage: AA(K_sage.gen()) # optional - flipper
1.122462048309373?
"""
r = K.lmbda.interval()
l = r.lower * ZZ(10)**(-r.precision)
u = r.upper * ZZ(10)**(-r.precision)
p = QQ['x'](K.coefficients)
s = AA.polynomial_root(p, RIF(l, u))
return NumberField(p, name, embedding=s)
def number_field_to_AA(a):
r"""
It is a mess to convert an element of a number field to the algebraic field
``AA``. This is a temporary fix.
"""
try:
return AA(a)
except TypeError:
return AA.polynomial_root(a.minpoly(), RIF(a))
def is_elliptic(self):
r"""
Return whether ``self`` is an elliptic matrix.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=7)
sage: [ G.V(k).is_elliptic() for k in range(1,8) ]
[False, False, False, False, False, False, True]
sage: G.U().is_elliptic()
True
"""
return AA(self.discriminant()) < 0
def __init__(self, lambda_squared=None, field=None):
if lambda_squared == None:
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
R = PolynomialRing(ZZ, 'x')
x = R.gen()
field = NumberField(x**3 - ZZ(5) * x**2 + ZZ(4) * x - ZZ(1),
'r',
embedding=AA(ZZ(4)))
self._l = field.gen()
else:
if field is None:
self._l = lambda_squared
field = lambda_squared.parent()
else:
self._l = field(lambda_squared)
Surface.__init__(self, field, ZZ.zero(), finite=False)
def lam_minpoly(self):
r"""
Return the minimal polynomial of the corresponding lambda parameter of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(10).lam_minpoly()
x^4 - 5*x^2 + 5
sage: HeckeTriangleGroup(17).lam_minpoly()
x^8 - x^7 - 7*x^6 + 6*x^5 + 15*x^4 - 10*x^3 - 10*x^2 + 4*x + 1
sage: HeckeTriangleGroup(infinity).lam_minpoly()
x - 2
"""
# TODO: Write an explicit (faster) implementation
lam_symbolic = 2*cos(pi/self._n)
return AA(lam_symbolic).minpoly()
def __init__(self, n):
r"""
Hecke triangle group (2, n, infinity).
Namely the von Dyck group corresponding to the triangle group
with angles (pi/2, pi/n, 0).
INPUT:
- ``n`` - ``infinity`` or an integer greater or equal to ``3``.
OUTPUT:
The Hecke triangle group for the given parameter ``n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(12)
sage: G
Hecke triangle group for n = 12
sage: G.category()
Category of groups
"""
self._n = n
if n in [3, infinity]:
self._base_ring = ZZ
self._lam = ZZ(1) if n==3 else ZZ(2)
else:
lam_symbolic = 2*cos(pi/n)
K = NumberField(self.lam_minpoly(), 'lam', embedding = AA(lam_symbolic))
#self._base_ring = K.order(K.gens())
self._base_ring = K.maximal_order()
self._lam = self._base_ring.gen(1)
T = matrix(self._base_ring, [[1,self._lam],[0,1]])
S = matrix(self._base_ring, [[0,-1],[1,0]])
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), self._base_ring, [S, T])
def homothety_rotation_decomposition(m):
r"""
Return a couple composed of the homothety and a rotation matrix.
The coefficients of the returned pair are either in the ground field of
``m`` or in the algebraic field ``AA``.
EXAMPLES::
sage: from flatsurf.geometry.matrix_2x2 import homothety_rotation_decomposition
sage: R.<x> = PolynomialRing(QQ)
sage: K.<sqrt2> = NumberField(x^2 - 2, embedding=1.4142)
sage: m = matrix([[sqrt2, -sqrt2],[sqrt2,sqrt2]])
sage: a,rot = homothety_rotation_decomposition(m)
sage: a
2
sage: rot
[ 1/2*sqrt2 -1/2*sqrt2]
[ 1/2*sqrt2 1/2*sqrt2]
"""
if not is_similarity(m):
raise ValueError("the matrix must be a similarity")
det = m.det()
if not det.is_square():
if not AA.has_coerce_map_from(m.base_ring()):
l = map(number_field_to_AA, m.list())
M = MatrixSpace(AA, 2)
m = M(l)
else:
m = m.change_ring(AA)
sqrt_det = det.sqrt()
return sqrt_det, m / sqrt_det
def homothety_rotation_decomposition(m):
r"""
Return a couple composed of the homothety and a rotation matrix.
The coefficients of the returned pair are either in the ground field of
``m`` or in the algebraic field ``AA``.
EXAMPLES::
sage: from flatsurf.geometry.matrix_2x2 import homothety_rotation_decomposition
sage: R.<x> = PolynomialRing(QQ)
sage: K.<sqrt2> = NumberField(x^2 - 2, embedding=1.4142)
sage: m = matrix([[sqrt2, -sqrt2],[sqrt2,sqrt2]])
sage: a,rot = homothety_rotation_decomposition(m)
sage: a
2
sage: rot
[ 1/2*sqrt2 -1/2*sqrt2]
[ 1/2*sqrt2 1/2*sqrt2]
"""
if not is_similarity(m):
raise ValueError("the matrix must be a similarity")
det = m.det()
if not det.is_square():
if not AA.has_coerce_map_from(m.base_ring()):
l = map(number_field_to_AA,m.list())
M = MatrixSpace(AA,2)
m = M(l)
else:
m = m.change_ring(AA)
sqrt_det = det.sqrt()
return sqrt_det, m / sqrt_det
def mcmullen_genus2_prototype(w, h, t, e, rel=0):
r"""
McMullen prototypes in the stratum H(2).
These prototype appear at least in McMullen "Teichmüller curves in genus
two: Discriminant and spin" (2004). The notation from that paper are
quadruple ``(a, b, c, e)`` which translates in our notation as
``w = b``, ``h = c``, ``t = a`` (and ``e = e``).
The associated discriminant is `D = e^2 + 4 wh`.
If ``rel`` is a positive parameter (less than w-lambda) the surface belongs
to the eigenform locus in H(1,1).
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from surface_dynamics import AbelianStratum
sage: prototypes = {
....: 5: [(1,1,0,-1)],
....: 8: [(1,1,0,-2), (2,1,0,0)],
....: 9: [(2,1,0,-1)],
....: 12: [(1,2,0,-2), (2,1,0,-2), (3,1,0,0)],
....: 13: [(1,1,0,-3), (3,1,0,-1), (3,1,0,1)],
....: 16: [(3,1,0,-2), (4,1,0,0)],
....: 17: [(1,2,0,-3), (2,1,0,-3), (2,2,0,-1), (2,2,1,-1), (4,1,0,-1), (4,1,0,1)],
....: 20: [(1,1,0,-4), (2,2,1,-2), (4,1,0,-2), (4,1,0,2)],
....: 21: [(1,3,0,-3), (3,1,0,-3)],
....: 24: [(1,2,0,-4), (2,1,0,-4), (3,2,0,0)],
....: 25: [(2,2,0,-3), (2,2,1,-3), (3,2,0,-1), (4,1,0,-3)]}
sage: for D in sorted(prototypes):
....: for w,h,t,e in prototypes[D]:
....: T = translation_surfaces.mcmullen_genus2_prototype(w,h,t,e)
....: assert T.stratum() == AbelianStratum(2)
....: assert (D.is_square() and T.base_ring() is QQ) or (T.base_ring().polynomial().discriminant() == D)
An example with some relative homology::
sage: U8 = translation_surfaces.mcmullen_genus2_prototype(2,1,0,0,1/4) # discriminant 8
sage: U12 = translation_surfaces.mcmullen_genus2_prototype(3,1,0,0,3/10) # discriminant 12
sage: U8.stratum()
H_2(1^2)
sage: U8.base_ring().polynomial().discriminant()
8
sage: U8.j_invariant()
(
[4 0]
(0), (0), [0 2]
)
sage: U12.stratum()
H_2(1^2)
sage: U12.base_ring().polynomial().discriminant()
12
sage: U12.j_invariant()
(
[6 0]
(0), (0), [0 2]
)
"""
w = ZZ(w)
h = ZZ(h)
t = ZZ(t)
e = ZZ(e)
g = w.gcd(h)
gg = g.gcd(t).gcd(e)
if w <= 0 or h <= 0 or t < 0 or t >= g or not g.gcd(t).gcd(
e).is_one() or e + h >= w:
raise ValueError("invalid parameters")
x = polygen(QQ)
poly = x**2 - e * x - w * h
if poly.is_irreducible():
emb = AA.polynomial_root(poly, RIF(0, w))
K = NumberField(poly, 'l', embedding=emb)
l = K.gen()
else:
K = QQ
D = e**2 + 4 * w * h
d = D.sqrt()
l = (e + d) / 2
rel = K(rel)
# (lambda,lambda) square on top
# twisted (w,0), (t,h)
s = Surface_list(base_ring=K)
if rel:
if rel < 0 or rel > w - l:
raise ValueError("invalid rel argument")
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (l + rel, l), (rel, l)],
ring=K))
s.add_polygon(
polygons(vertices=[(0, 0), (rel, 0), (rel + l, 0), (w, 0),
(w + t, h), (l + rel + t, h), (t + l, h),
(t, h)],
ring=K))
s.set_edge_pairing(0, 1, 0, 3)
s.set_edge_pairing(0, 0, 1, 6)
s.set_edge_pairing(0, 2, 1, 1)
s.set_edge_pairing(1, 2, 1, 4)
s.set_edge_pairing(1, 3, 1, 7)
s.set_edge_pairing(1, 0, 1, 5)
else:
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (l, l), (0, l)], ring=K))
s.add_polygon(
polygons(vertices=[(0, 0), (l, 0), (w, 0), (w + t, h),
(l + t, h), (t, h)],
ring=K))
s.set_edge_pairing(0, 1, 0, 3)
s.set_edge_pairing(0, 0, 1, 4)
s.set_edge_pairing(0, 2, 1, 0)
s.set_edge_pairing(1, 1, 1, 3)
s.set_edge_pairing(1, 2, 1, 5)
s.set_immutable()
return TranslationSurface(s)
def fixed_points(self, embedded=False, order="default"):
r"""
Return a pair of (mutually conjugate) fixed points of ``self``
in a possible quadratic extension of the base field.
INPUT:
- ``embedded`` -- If ``True`` the fixed points are embedded into
``AlgebraicRealField`` resp. ``AlgebraicField``.
Default: ``False``.
- ``order`` -- If ``order="none"`` the fixed points are choosen
and ordered according to a fixed formula.
If ``order="sign"`` the fixed points are always ordered
according to the sign in front of the square root.
If ``order="default"`` (default) then in case the fixed
points are hyperbolic they are ordered according to the
sign of the trace of ``self`` instead, such that the
attracting fixed point comes first.
If ``order="trace"`` the fixed points are always ordered
according to the sign of the trace of ``self``.
If the trace is zero they are ordered by the sign in
front of the square root. In particular the fixed_points
in this case remain the same for ``-self``.
OUTPUT:
If ``embedded=True`` an element of either ``AlgebraicRealField`` or
``AlgebraicField`` is returned. Otherwise an element of a relative field
extension over the base field of (the parent of) ``self`` is returned.
Warning: Relative field extensions don't support default embeddings.
So the correct embedding (which is the positive resp. imaginary positive
one) has to be choosen.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=infinity)
sage: (-G.T(-4)).fixed_points()
[+Infinity, +Infinity]
sage: (-G.S()).fixed_points()
[1/2*e, -1/2*e]
sage: p = (-G.S()).fixed_points(embedded=True)[0]
sage: p
1*I
sage: (-G.S()).acton(p) == p
True
sage: (-G.V(2)).fixed_points()
[1/2*e, -1/2*e]
sage: (-G.V(2)).fixed_points() == G.V(2).fixed_points()
True
sage: p = (-G.V(2)).fixed_points(embedded=True)[1]
sage: p
-1.732050807568878?
sage: (-G.V(2)).acton(p) == p
True
sage: G = HeckeTriangleGroup(n=7)
sage: (-G.S()).fixed_points()
[1/2*e, -1/2*e]
sage: p = (-G.S()).fixed_points(embedded=True)[1]
sage: p
-1*I
sage: (-G.S()).acton(p) == p
True
sage: (G.U()^4).fixed_points()
[(1/2*lam^2 - 1/2*lam - 1/2)*e + 1/2*lam, (-1/2*lam^2 + 1/2*lam + 1/2)*e + 1/2*lam]
sage: pts = (G.U()^4).fixed_points(order="trace")
sage: (G.U()^4).fixed_points() == [pts[1], pts[0]]
True
sage: (G.U()^4).fixed_points(order="trace") == (-G.U()^4).fixed_points(order="trace")
True
sage: (G.U()^4).fixed_points() == (G.U()^4).fixed_points(order="none")
True
sage: (-G.U()^4).fixed_points() == (G.U()^4).fixed_points()
True
sage: (-G.U()^4).fixed_points(order="none") == pts
True
sage: p = (G.U()^4).fixed_points(embedded=True)[1]
sage: p
0.9009688679024191? - 0.4338837391175581?*I
sage: (G.U()^4).acton(p) == p
True
sage: (-G.V(5)).fixed_points()
[(1/2*lam^2 - 1/2*lam - 1/2)*e, (-1/2*lam^2 + 1/2*lam + 1/2)*e]
sage: (-G.V(5)).fixed_points() == G.V(5).fixed_points()
True
sage: p = (-G.V(5)).fixed_points(embedded=True)[0]
sage: p
0.6671145837954892?
sage: (-G.V(5)).acton(p) == p
True
"""
if self.c() == 0:
return [infinity, infinity]
else:
D = self.discriminant()
if D.is_square():
e = D.sqrt()
else:
e = self.root_extension_field().gen()
a = self.a()
d = self.d()
c = self.c()
if order == "none":
sgn = ZZ(1)
elif order == "sign":
sgn = AA(c).sign()
elif order == "default":
if self.is_elliptic() or self.trace() == 0:
sgn = AA(c).sign()
else:
sgn = AA(self.trace()).sign()
elif order == "trace":
if self.trace() == 0:
sgn = AA(c).sign()
else:
sgn = AA(self.trace()).sign()
else:
raise NotImplementedError
if embedded:
e = AA(D).sqrt()
a = AA(a)
d = AA(d)
c = AA(c)
root1 = (a - d) / (2 * c) + sgn * e / (2 * c)
root2 = (a - d) / (2 * c) - sgn * e / (2 * c)
return [root1, root2]
def decompose_basic(self):
r"""
Decompose ``self`` into a product of the generators
``S`` and ``T`` of its parent,
The function returns a tuple ``L`` consisting of either
``parent.S()`` or non-trivial integer powers of ``parent.T()``.
Additionally ``L`` starts with +- the identity.
It satisfies the property: ``prod(L) == self``.
If this decomposition is not possible an ``ValueError``
is raised. In particular this function can be used to
check the membership in ``parent`` of an arbitrary matrix
over the base ring.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=17)
sage: L = (-G.V(2)).decompose_basic()
sage: L
(
[ 1 lam] [ 0 -1] [ 1 lam] [-1 0]
[ 0 1], [ 1 0], [ 0 1], [ 0 -1]
)
sage: prod(L) == -G.V(2)
True
sage: L = G.U().decompose_basic()
sage: L
(
[ 1 lam] [ 0 -1] [1 0]
[ 0 1], [ 1 0], [0 1]
)
sage: prod(L) == G.U()
True
"""
def mshift(A):
a = A[0][0]
b = A[0][1]
c = A[1][0]
d = A[1][1]
return (4 * a * c + b * d) / (4 * c * c + d * d)
def mabs(A):
a = A[0][0]
b = A[0][1]
c = A[1][0]
d = A[1][1]
return (4 * a * a + b * b) / (4 * c * c + d * d)
res = []
T = self.parent().T()
S = self.parent().S()
M = self
lam = self.parent().lam()
while True:
m = (AA(mshift(M) / lam) + ZZ(1) / ZZ(2)).floor()
M = T**(-m) * M
if (m != 0):
res.append(T**m)
abs_t = mabs(M)
if AA(abs_t) < 1:
M = (-S) * M
res.append(S)
elif M.is_identity():
res.append(M)
return tuple(res)
else:
raise ValueError(
"The matrix is not an element of {}, up to equivalence it identifies two nonequivalent points."
.format(self.parent()))
def root_extension_embedding(self, K=AlgebraicField()):
r"""
Return the correct embedding from the root extension field to ``K`` (default: ``AlgebraicField()``).
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(n=infinity)
sage: fp = (-G.S()).fixed_points()[0]
sage: alg_fp = (-G.S()).root_extension_embedding()(fp)
sage: alg_fp
1*I
sage: alg_fp == (-G.S()).fixed_points(embedded=True)[0]
True
sage: fp = (-G.V(2)).fixed_points()[1]
sage: alg_fp = (-G.V(2)).root_extension_embedding(AA)(fp)
sage: alg_fp
-1.732050807568...?
sage: alg_fp == (-G.V(2)).fixed_points(embedded=True)[1]
True
sage: fp = (-G.V(2)).fixed_points()[0]
sage: alg_fp = (-G.V(2)).root_extension_embedding(AA)(fp)
sage: alg_fp
1.732050807568...?
sage: alg_fp == (-G.V(2)).fixed_points(embedded=True)[0]
True
sage: G = HeckeTriangleGroup(n=7)
sage: fp = (-G.S()).fixed_points()[1]
sage: alg_fp = (-G.S()).root_extension_embedding()(fp)
sage: alg_fp
0.?... - 1.000000000000...?*I
sage: alg_fp == (-G.S()).fixed_points(embedded=True)[1]
True
sage: fp = (-G.U()^4).fixed_points()[0]
sage: alg_fp = (-G.U()^4).root_extension_embedding()(fp)
sage: alg_fp
0.9009688679024...? + 0.4338837391175...?*I
sage: alg_fp == (-G.U()^4).fixed_points(embedded=True)[0]
True
sage: (-G.U()^4).root_extension_embedding(CC)(fp)
0.900968867902... + 0.433883739117...*I
sage: (-G.U()^4).root_extension_embedding(CC)(fp).parent()
Complex Field with 53 bits of precision
sage: fp = (-G.V(5)).fixed_points()[1]
sage: alg_fp = (-G.V(5)).root_extension_embedding(AA)(fp)
sage: alg_fp
-0.6671145837954...?
sage: alg_fp == (-G.V(5)).fixed_points(embedded=True)[1]
True
"""
G = self.parent()
D = self.discriminant()
F = self.root_extension_field()
n = G.n()
if D.is_square():
e = D.sqrt()
lam = G.lam()
elif n in [3, infinity]:
e = F.gen(0)
lam = G.lam()
else:
(e, lam) = F.gens()
emb_lam = K(G.lam())
if self.is_elliptic():
emb_e = K(AlgebraicField()(D).sqrt())
else:
emb_e = K(AA(D).sqrt())
guess = ZZ(0)
min_value = infinity
index = ZZ(0)
for (index,
emb) in enumerate(self.root_extension_field().embeddings(K)):
if K.is_exact():
if emb(lam) == emb_lam and emb(e) == emb_e:
return emb
else:
value = (emb(lam) - emb_lam).n(
K.prec()).abs() + (emb(e) - emb_e).n(K.prec()).abs()
if (value < min_value):
guess = index
min_value = value
if K.is_exact() or min_value == infinity:
raise ValueError(
"No suitable embedding is available for K = {}!".format(K))
else:
return self.root_extension_field().embeddings(K)[guess]
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp_translation_surface(ss)==0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g=ZZ(genus)
assert g>=3
from sage.rings.polynomial.polynomial_ring import polygen
x = polygen(AA)
p=sum([x**i for i in xrange(1,g+1)])-1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1/2, 1))
field=NumberField(alpha_AA.minpoly(),'alpha',embedding=alpha_AA)
a=field.gen()
from sage.modules.free_module import VectorSpace
V=VectorSpace(field,2)
p=[None for i in xrange(g+1)]
q=[None for i in xrange(g+1)]
p[0]=V(( (1-a**g)/2, a**2/(1-a) ))
q[0]=V(( -a**g/2, a ))
p[1]=V(( -(a**(g-1)+a**g)/2, (a-a**2+a**3)/(1-a) ))
p[g]=V(( 1+(a-a**g)/2, (3*a-1-a**2)/(1-a) ))
for i in xrange(2,g):
p[i]=V(( (a-a**i)/(1-a) , a/(1-a) ))
for i in xrange(1,g+1):
q[i]=V(( (2*a-a**i-a**(i+1))/(2*(1-a)), (a-a**(g-i+2))/(1-a) ))
from flatsurf.geometry.polygon import Polygons
P=Polygons(field)
s = Surface_list(field)
T = [None] * (2*g+1)
Tp = [None] * (2*g+1)
from sage.matrix.constructor import Matrix
m=Matrix([[1,0],[0,-1]])
for i in xrange(1,g+1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i]=s.add_polygon(P(edges=[ q[i]-p[0], q[i-1]-q[i], p[0]-q[i-1] ]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g+i]=s.add_polygon(P(edges=[ q[i-1]-p[i], q[i]-q[i-1], p[i]-q[i] ]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i]=s.add_polygon(m*s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g+i]=s.add_polygon(m*s.polygon(T[g+i]))
for i in xrange(1,g):
s.change_edge_gluing(T[i],0,T[i+1],2)
s.change_edge_gluing(Tp[i],2,Tp[i+1],0)
for i in xrange(1,g+1):
s.change_edge_gluing(T[i],1,T[g+i],1)
s.change_edge_gluing(Tp[i],1,Tp[g+i],1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1],2,Tp[g],2)
s.change_edge_gluing(Tp[1],0,T[g],0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g+1],2,Tp[2*g],2)
s.change_edge_gluing(Tp[g+1],0,T[2*g],0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g+1],0,T[2*g-1],2)
s.change_edge_gluing(Tp[g+1],2,Tp[2*g-1],0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2*g],2,T[g+2],0)
s.change_edge_gluing(Tp[2*g],0,Tp[g+2],2)
for i in xrange(2,g-1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g+i],2,Tp[g+i+1],2)
s.change_edge_gluing(Tp[g+i],0,T[g+i+1],0)
s.set_immutable()
return TranslationSurface(s)
def _coerce_map_from_(self, P):
r"""
Return whether ``P`` coerces into this symbolic subring.
INPUT:
- ``P`` -- a parent.
OUTPUT:
A boolean or ``None``.
TESTS::
sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_coerce_map_from(SR) # indirect doctest # not tested see #19231
False
::
sage: from sage.symbolic.subring import SymbolicSubring
sage: C = SymbolicSubring(no_variables=True)
sage: C.has_coerce_map_from(ZZ) # indirect doctest
True
sage: C.has_coerce_map_from(QQ) # indirect doctest
True
sage: C.has_coerce_map_from(RR) # indirect doctest
True
sage: C.has_coerce_map_from(RIF) # indirect doctest
True
sage: C.has_coerce_map_from(CC) # indirect doctest
True
sage: C.has_coerce_map_from(CIF) # indirect doctest
True
sage: C.has_coerce_map_from(AA) # indirect doctest
True
sage: C.has_coerce_map_from(QQbar) # indirect doctest
True
sage: C.has_coerce_map_from(SR) # indirect doctest
False
"""
if P == SR:
# Workaround; can be deleted once #19231 is fixed
return False
from sage.rings.real_mpfr import mpfr_prec_min
from sage.rings.all import (ComplexField,
RLF, CLF, AA, QQbar, InfinityRing)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
if isinstance(P, type):
return SR._coerce_map_from_(P)
elif RLF.has_coerce_map_from(P) or \
CLF.has_coerce_map_from(P) or \
AA.has_coerce_map_from(P) or \
QQbar.has_coerce_map_from(P):
return True
elif (P is InfinityRing or
is_RealIntervalField(P) or is_ComplexIntervalField(P)):
return True
elif ComplexField(mpfr_prec_min()).has_coerce_map_from(P):
return P not in (RLF, CLF, AA, QQbar)
def angle(u, v, numerical=False, assume_rational=False):
r"""
Return the angle between the vectors ``u`` and ``v`` divided by `2 \pi`.
INPUT:
- ``u``, ``v`` - vectors
- ``numerical`` - boolean, whether to return floating point numbers
- ``assume_rational`` - whether we assume that the angle is a multiple
rational of ``pi``. By default it is ``False`` but if it is known in
advance that the result is rational then setting it to ``True`` might be
much faster.
EXAMPLES::
sage: from flatsurf.geometry.matrix_2x2 import angle
As the implementation is dirty, we at least check that it works for all
denominator up to 20::
sage: u = vector((AA(1),AA(0)))
sage: for n in xsrange(1,20): # long time (10 sec)
....: for k in xsrange(1,n):
....: v = vector((AA(cos(2*k*pi/n)), AA(sin(2*k*pi/n))))
....: assert angle(u,v) == k/n
The numerical version (working over floating point numbers)::
sage: import math
sage: u = (1, 0)
sage: for n in xsrange(1,20):
....: for k in xsrange(1,n):
....: a = 2 * k * math.pi / n
....: v = (math.cos(a), math.sin(a))
....: assert abs(angle(u,v,numerical=True) * 2 * math.pi - a) < 1.e-10
And we test up to 50 when setting ``assume_rational`` to ``True``::
sage: for n in xsrange(1,20): # long time
....: for k in xsrange(1,n):
....: v = vector((AA(cos(2*k*pi/n)), AA(sin(2*k*pi/n))))
....: assert angle(u,v,assume_rational=True) == k/n
If the angle is not rational, then the method returns an element in the real
lazy field::
sage: v = vector((AA(sqrt(2)), AA(sqrt(3))))
sage: a = angle(u, v)
sage: a # abs tol 1e-14
0.14102355421224375
sage: exp(2*pi.n()*CC(0,1)*a)
0.632455532033676 + 0.774596669241483*I
sage: v / v.norm()
(0.6324555320336758?, 0.774596669241484?)
"""
if not assume_rational and not numerical:
sqnorm_u = u[0] * u[0] + u[1] * u[1]
sqnorm_v = v[0] * v[0] + v[1] * v[1]
if sqnorm_u != sqnorm_v:
uu = vector(AA, u)
vv = (AA(sqnorm_u) / AA(sqnorm_v)).sqrt() * vector(AA, v)
else:
uu = u
vv = v
cos_uv = (uu[0] * vv[0] + uu[1] * vv[1]) / sqnorm_u
sin_uv = (uu[0] * vv[1] - uu[1] * vv[0]) / sqnorm_u
is_rational = is_cosine_sine_of_rational(cos_uv, sin_uv)
elif assume_rational:
is_rational = True
import math
u0 = float(u[0])
u1 = float(u[1])
v0 = float(v[0])
v1 = float(v[1])
cos_uv = (u0 * v0 + u1 * v1) / math.sqrt(
(u0 * u0 + u1 * u1) * (v0 * v0 + v1 * v1))
if cos_uv < -1.0:
assert cos_uv > -1.0000001
cos_uv = -1.0
elif cos_uv > 1.0:
assert cos_uv < 1.0000001
cos_uv = 1.0
angle = math.acos(cos_uv) / (2 * math.pi) # rat number between 0 and 1/2
if numerical or not is_rational:
return 1.0 - angle if u0 * v1 - u1 * v0 < 0 else angle
else:
# fast and dirty way using floating point approximation
# (see below for a slow but exact method)
angle_rat = RR(angle).nearby_rational(0.00000001)
if angle_rat.denominator() > 100:
raise NotImplementedError(
"the numerical method used is not smart enough!")
return 1 - angle_rat if u0 * v1 - u1 * v0 < 0 else angle_rat
- 251568270025211201955389171860338579171675714868104052152789880989805712726614197574887246767925108246607970007160940134370673529705101398997875679835167365081669374673539993436160000*t**3
- 141772430429024326881509616736177148212059744813752208473666088996700368688933789363560771135161864201721524861419389136852260702560567771393655430337183040528733205980885277265100800000*t**2
- 25817606346086956476758470208850883766420383787607030339022713274189365873405154695767797745935894060883806800961601773820707673481632102376182116930107064769426021006200320503617211596800*t
+ 34918206405098505823938790072675572231488655998026261577866691497637535623661745400444768148106948876570405151317417742685700849593772165309178580940805695949542187927076864000)*Dt)
quadric_slice_pol = (
4980990673427087034113103774848375913397675011396681161337606780457883155824640000000000*t**12
- 16313074573215242896867677175985719375664055250377801991087546344967331905536000000000*t**9
- 14852779293587242300314544658084523021409425155052443959294262319432698489552764928000000*t**8
+ 18694126910889886952945780127491545129704079293214429569400282861674612412907520000*t**6
+ 32429224374768702788524801575483580065598417846595577296275963028007688596147404800000*t**5
+ 14763130935033327878568955564665179022508855828282305094488782847988800598441515915673600*t**4
- 7447056930374930458107131157447569387299331973073657492405996702806537404416000*t**3
- 18581243794708202636835504417848386599346688512251081679746508518773002589362454528*t**2
- 16116744082275656666424675660780874575937043631040306492377025123023286892432343685120*t
- 4891341219838850087826096307272910719484535278552470341569283855964428449539674077056375)
quadric_slice_crit = AA.polynomial_root(quadric_slice_pol, RIF(-0.999,-0.998))
aa = AA.polynomial_root(AA.common_polynomial(t**2 - t - 6256320), RIF(-RR(2500.7637305969961), -RR(2500.7637305969956)))
K, a = NumberField(t**2 - t - 6256320, 'a', embedding=aa).objgen()
DiffOps_x, x, Dx = DifferentialOperators(K, 'x')
iint_quadratic_alg = IVP(
dop = (
(8680468749131953125000000000000000000000*x**13
+ (34722222218750000000000000000000*a
- 8680555572048611109375000000000000000000)*x**12
- 43419899820094632213834375000000000000000*x**11
+ (
-173681336093739466250000000000000*a
+ 43420334110275534609369733125000000000000)*x**10
+ 86874920665761352031076792873375000000000*x**9
+ (347503157694622354347850650000000*a
def _coerce_map_from_(self, P):
r"""
Return whether ``P`` coerces into this symbolic subring.
INPUT:
- ``P`` -- a parent.
OUTPUT:
A boolean or ``None``.
TESTS::
sage: from sage.symbolic.subring import GenericSymbolicSubring
sage: GenericSymbolicSubring(vars=tuple()).has_coerce_map_from(SR) # indirect doctest # not tested see #19231
False
::
sage: from sage.symbolic.subring import SymbolicSubring
sage: C = SymbolicSubring(no_variables=True)
sage: C.has_coerce_map_from(ZZ) # indirect doctest
True
sage: C.has_coerce_map_from(QQ) # indirect doctest
True
sage: C.has_coerce_map_from(RR) # indirect doctest
True
sage: C.has_coerce_map_from(RIF) # indirect doctest
True
sage: C.has_coerce_map_from(CC) # indirect doctest
True
sage: C.has_coerce_map_from(CIF) # indirect doctest
True
sage: C.has_coerce_map_from(AA) # indirect doctest
True
sage: C.has_coerce_map_from(QQbar) # indirect doctest
True
sage: C.has_coerce_map_from(SR) # indirect doctest
False
"""
if P == SR:
# Workaround; can be deleted once #19231 is fixed
return False
from sage.rings.real_mpfr import mpfr_prec_min
from sage.rings.all import (ComplexField, RLF, CLF, AA, QQbar,
InfinityRing)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
if isinstance(P, type):
return SR._coerce_map_from_(P)
elif RLF.has_coerce_map_from(P) or \
CLF.has_coerce_map_from(P) or \
AA.has_coerce_map_from(P) or \
QQbar.has_coerce_map_from(P):
return True
elif (P is InfinityRing or is_RealIntervalField(P)
or is_ComplexIntervalField(P)):
return True
elif ComplexField(mpfr_prec_min()).has_coerce_map_from(P):
return P not in (RLF, CLF, AA, QQbar)
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp(ss) == 0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g = ZZ(genus)
assert g >= 3
x = polygen(AA)
p = sum([x**i for i in range(1, g + 1)]) - 1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1 / 2, 1))
field = NumberField(alpha_AA.minpoly(), 'alpha', embedding=alpha_AA)
a = field.gen()
V = VectorSpace(field, 2)
p = [None for i in range(g + 1)]
q = [None for i in range(g + 1)]
p[0] = V(((1 - a**g) / 2, a**2 / (1 - a)))
q[0] = V((-a**g / 2, a))
p[1] = V((-(a**(g - 1) + a**g) / 2, (a - a**2 + a**3) / (1 - a)))
p[g] = V((1 + (a - a**g) / 2, (3 * a - 1 - a**2) / (1 - a)))
for i in range(2, g):
p[i] = V(((a - a**i) / (1 - a), a / (1 - a)))
for i in range(1, g + 1):
q[i] = V(((2 * a - a**i - a**(i + 1)) / (2 * (1 - a)),
(a - a**(g - i + 2)) / (1 - a)))
P = ConvexPolygons(field)
s = Surface_list(field)
T = [None] * (2 * g + 1)
Tp = [None] * (2 * g + 1)
from sage.matrix.constructor import Matrix
m = Matrix([[1, 0], [0, -1]])
for i in range(1, g + 1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i] = s.add_polygon(
P(edges=[q[i] - p[0], q[i - 1] - q[i], p[0] - q[i - 1]]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g + i] = s.add_polygon(
P(edges=[q[i - 1] - p[i], q[i] - q[i - 1], p[i] - q[i]]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i] = s.add_polygon(m * s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g + i] = s.add_polygon(m * s.polygon(T[g + i]))
for i in range(1, g):
s.change_edge_gluing(T[i], 0, T[i + 1], 2)
s.change_edge_gluing(Tp[i], 2, Tp[i + 1], 0)
for i in range(1, g + 1):
s.change_edge_gluing(T[i], 1, T[g + i], 1)
s.change_edge_gluing(Tp[i], 1, Tp[g + i], 1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1], 2, Tp[g], 2)
s.change_edge_gluing(Tp[1], 0, T[g], 0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g + 1], 2, Tp[2 * g], 2)
s.change_edge_gluing(Tp[g + 1], 0, T[2 * g], 0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g + 1], 0, T[2 * g - 1], 2)
s.change_edge_gluing(Tp[g + 1], 2, Tp[2 * g - 1], 0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2 * g], 2, T[g + 2], 0)
s.change_edge_gluing(Tp[2 * g], 0, Tp[g + 2], 2)
for i in range(2, g - 1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g + i], 2, Tp[g + i + 1], 2)
s.change_edge_gluing(Tp[g + i], 0, T[g + i + 1], 0)
s.set_immutable()
return TranslationSurface(s)