def min_reol_maybe_with3gens(data):
forms = data.relatively_prime_3forms_maybe()
d = data.brackets_dict
if forms is None:
return None
F = FreeModule(R, 2)
f, g, h = forms
e0, e1 = F.gens()
a = d[(g, h)]
b = -d[(f, h)]
c = d[(f, g)]
f = e0 * c
g = e1 * c
h = -(a * e0 + b * e1)
if c.degree() > 0:
n = smodule(f, h)
idl = sideal(b)
m = squotient(n, idl)
else:
m = smodule(f, g)
wts = data.weight_of_basis()
mls = list(takewhile(lambda l: any(x != 0 for x in l), slist(smres(m, 0))))
mls = [[list(v) for v in list(l)] for l in mls]
wts_of_mls = []
wts_of_mls.append([degree_vec(v, wts) for v in mls[0]])
for i, l in enumerate(mls[1:]):
wts = wts_of_mls[i]
wts_of_mls.append([degree_vec(v, wts) for v in l])
return (mls, wts_of_mls, (forms[0], forms[1], c))
def __init__(self, base_ring, coordinate):
self.coordinate = coordinate
self.Vector = flatsurf.Vector._unwrapped[self.coordinate]
self._isomorphic_vector_space = FreeModule(base_ring, 2)
if isinstance(base_ring, real_embedded_number_field.RealEmbeddedNumberField):
self._isomorphic_vector_space = FreeModule(base_ring.number_field, 2)
Parent.__init__(self, base_ring, category=FreeModules(base_ring))
self.register_coercion(self._isomorphic_vector_space)
self._isomorphic_vector_space.register_conversion(ConversionVectorSpace(self))
def is_product(n, den_tuple):
r"""
INPUT:
- ``n`` - number of variables
- ``den_tuple`` - tuple of pairs ``(vector, power)``
TESTS::
sage: from surface_dynamics.misc.generalized_multiple_zeta_values import is_product
sage: is_product(3, [((1,0,0),2), ((0,1,0),5), ((1,1,0),1), ((0,0,1),3)])
[(2, (((0, 1), 5), ((1, 0), 2), ((1, 1), 1))), (1, (((1), 3),))]
sage: is_product(3, [((1,0,0),2), ((0,1,0),3), ((1,0,1),1), ((0,0,1),5)])
[(2, (((0, 1), 5), ((1, 0), 2), ((1, 1), 1))), (1, (((1), 3),))]
sage: is_product(3, [((1,1,1),3)]) is None
True
"""
D = DisjointSet(n)
assert all(len(v) == n for v, p in den_tuple), (n, den_tuple)
# 1. product structure
for v, _ in den_tuple:
i0 = 0
while not v[i0]:
i0 += 1
i = i0 + 1
while i < n:
if v[i]:
D.union(i0, i)
i += 1
if D.number_of_subsets() == 1:
# no way to split variables
return
# split variables
Rdict = D.root_to_elements_dict()
keys = sorted(Rdict.keys())
key_indices = {k: i for i, k in enumerate(keys)}
values = [Rdict[k] for k in keys]
values_indices = [{v: i for i, v in enumerate(v)} for v in values]
n_list = [len(J) for J in values]
F = [FreeModule(ZZ, nn) for nn in n_list]
new_terms = [[] for _ in range(len(Rdict))]
for v, p in den_tuple:
i0 = 0
while not v[i0]:
i0 += 1
i0 = D.find(i0)
assert all(D.find(i) == i0 for i in range(n)
if v[i]), (i0, [D.find(i) for i in range(n) if v[i]])
k = key_indices[i0]
vv = F[k]()
for i in range(n):
if v[i]:
vv[values_indices[k][i]] = v[i]
vv.set_immutable()
new_terms[k].append((vv, p))
return list(zip(n_list, [tuple(sorted(terms)) for terms in new_terms]))
def __init__(self, surface):
if isinstance(surface, TranslationSurface):
base_ring = surface.base_ring()
from flatsurf.geometry.pyflatsurf_conversion import to_pyflatsurf
self._surface = to_pyflatsurf(surface)
else:
from flatsurf.geometry.pyflatsurf_conversion import sage_base_ring
base_ring, _ = sage_base_ring(surface)
self._surface = surface
# A model of the vector space R² in libflatsurf, e.g., to represent the
# vector associated to a saddle connection.
self.V2 = pyflatsurf.vector.Vectors(base_ring)
# We construct a spanning set of edges, that is a subset of the
# edges that form a basis of H_1(S, Sigma; Z)
# It comes together with a projection matrix
t, m = self._spanning_tree()
assert set(t.keys()) == set(f[0] for f in self._faces())
self.spanning_set = []
v = set(t.values())
for e in self._surface.edges():
if e.positive() not in v and e.negative() not in v:
self.spanning_set.append(e)
self.d = len(self.spanning_set)
assert 3*self.d - 3 == self._surface.size()
assert m.rank() == self.d
m = m.transpose()
# projection matrix from Z^E to H_1(S, Sigma; Z) in the basis
# of spanning edges
self.proj = matrix(ZZ, [r for r in m.rows() if not r.is_zero()])
self.Omega = self._intersection_matrix(t, self.spanning_set)
self.V = FreeModule(self.V2.base_ring(), self.d)
self.H = matrix(self.V2.base_ring(), self.d, 2)
for i in range(self.d):
s = self._surface.fromHalfEdge(self.spanning_set[i].positive())
self.H[i] = self.V2._isomorphic_vector_space(self.V2(s))
self.Hdual = self.Omega * self.H
# Note that we don't use Sage vector spaces because they are usually
# way too slow (in particular we avoid calling .echelonize())
self._U = matrix(self.V2._algebraic_ring(), self.d)
if self._U.base_ring() is not QQ:
from sage.all import next_prime
self._good_prime = self._U.base_ring().ideal(next_prime(2**30)).factor()[0][0]
self._Ubar = matrix(self._good_prime.residue_field(), self.d)
self._U_rank = 0
self.update_tangent_space_from_vector(self.H.transpose()[0])
self.update_tangent_space_from_vector(self.H.transpose()[1])
def test_skipper_cpa_enc(skipper=Skipper4, kyber=Kyber, t=128, l=None, exhaustive=False):
if not exhaustive:
for i in range(t):
pk, sk = kyber.key_gen(seed=i)
m0 = random_vector(GF(2), kyber.n)
c = skipper.enc(kyber, pk, m0, seed=i, l=l)
m1 = kyber.dec(sk, c)
assert(m0 == m1)
else:
# exhaustive test
for i in range(16):
pk, sk = kyber.key_gen(seed=i)
for m0 in FreeModule(GF(2), kyber.n):
c = skipper.enc(kyber, pk, m0, seed=i, l=l)
m1 = kyber.dec(sk, c)
assert(m0 == m1)
def boundaries(self):
r"""
Return the list of boundaries (ie sum of edges around a triangular face).
These are elements of H_1(S, Sigma; Z).
TESTS::
sage: from flatsurf import polygons, similarity_surfaces
sage: from flatsurf import GL2ROrbitClosure # optional: pyflatsurf
sage: from itertools import product
sage: for a in range(1,5): # optional: pyflatsurf
....: for b in range(a, 5):
....: for c in range(b, 5):
....: if gcd([a, b, c]) != 1 or (a,b,c) == (1,1,2):
....: continue
....: T = polygons.triangle(a, b, c)
....: S = similarity_surfaces.billiard(T)
....: S = S.minimal_cover(cover_type="translation")
....: O = GL2ROrbitClosure(S)
....: for b in O.boundaries():
....: assert (O.proj * b).is_zero()
"""
n = self._surface.size()
V = FreeModule(ZZ, n)
B = []
for (f1,f2,f3) in self._faces():
i1 = f1.index()
s1 = -1 if i1%2 else 1
i2 = f2.index()
s2 = -1 if i2%2 else 1
i3 = f3.index()
s3 = -1 if i3%2 else 1
i1 = f1.edge().index()
i2 = f2.edge().index()
i3 = f3.edge().index()
v = [0] * n
v[i1] = 1
v[i2] = s1 * s2
v[i3] = s1 * s3
B.append(V(v))
B[-1].set_immutable()
return B
def linear_forms(d):
r"""
EXAMPLES::
sage: from surface_dynamics.misc.generalized_multiple_zeta_values import linear_forms
sage: list(linear_forms(1))
[(1)]
sage: list(linear_forms(2))
[(1, 0), (0, 1), (1, 1)]
sage: L1 = list(linear_forms(3))
sage: L2 = L1[:]
sage: L2.sort(key = lambda x: x[::-1])
sage: assert L1 == L2
"""
F = FreeModule(ZZ, d)
for n in range(1, 2**d):
v = F(ZZ(n).digits(2, padto=d))
v.set_immutable()
yield v
class Vectors(UniqueRepresentation, Parent):
r"""
The subspace of the vector space `\mathbb{R}^2` modeled by vectors with
entries in ``base_ring``. Unlike normal SageMath vector spaces, this space
is backed by flatsur's ``Vector`` class with entries in ``coordinate``.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: K = NumberField(x**2 - 2, 'a', embedding=sqrt(AA(2)))
sage: V = Vectors(K)
sage: v = V((1,K.gen() - 3/2))
sage: v
(1, (a-3/2 ~ -0.085786438))
sage: VectorSpace(K, 2)(v)
(1, a - 3/2)
sage: from pyeantic import RealEmbeddedNumberField
sage: R = RealEmbeddedNumberField(K)
sage: Vectors(R) is V
True
sage: Vectors(R.renf) is V
True
sage: from gmpxxyy import mpz, mpq
sage: Vectors(QQ) is Vectors(mpq)
True
sage: Vectors(ZZ) is Vectors(mpz)
True
"""
Element = Vector
@staticmethod
def __classcall__(cls, base_ring):
if base_ring in [ZZ, mpz]:
base_ring = ZZ
coordinate = mpz
elif base_ring in [QQ, mpq]:
base_ring = QQ
coordinate = mpq
elif isinstance(base_ring, real_embedded_number_field.RealEmbeddedNumberField):
coordinate = cppyy.gbl.eantic.renf_elem_class
elif isinstance(base_ring, eantic.renf_class):
base_ring = real_embedded_number_field.RealEmbeddedNumberField(base_ring)
coordinate = cppyy.gbl.eantic.renf_elem_class
elif isinstance(base_ring, SageNumberField):
base_ring = real_embedded_number_field.RealEmbeddedNumberField(base_ring)
coordinate = cppyy.gbl.eantic.renf_elem_class
elif isinstance(base_ring, ExactReals):
coordinate = base_ring._element_factory
else:
raise NotImplementedError("unsupported base_ring {!r}".format(base_ring))
return super(Vectors, cls).__classcall__(cls, base_ring, coordinate)
def __init__(self, base_ring, coordinate):
self.coordinate = coordinate
self.Vector = flatsurf.Vector._unwrapped[self.coordinate]
self._isomorphic_vector_space = FreeModule(base_ring, 2)
if isinstance(base_ring, real_embedded_number_field.RealEmbeddedNumberField):
self._isomorphic_vector_space = FreeModule(base_ring.number_field, 2)
Parent.__init__(self, base_ring, category=FreeModules(base_ring))
self.register_coercion(self._isomorphic_vector_space)
self._isomorphic_vector_space.register_conversion(ConversionVectorSpace(self))
def _algebraic_ring(self):
r"""
Return the Sage algebraic ring which underlies our base ring; typically
this is (an isomorphic ring to) the base ring itself but for
non-algebraic rings such as ``ExactReals``, this gives the coefficient
ring.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: from pyexactreal import ExactReals
sage: Vectors(QQ)._algebraic_ring()
Rational Field
sage: R = ExactReals()
sage: Vectors(R)._algebraic_ring()
Rational Field
sage: K = NumberField(x^3 - 2, 'a', embedding=AA(2)**(1/3))
sage: Vectors(K)._algebraic_ring()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
sage: R = ExactReals(K)
sage: Vectors(R)._algebraic_ring()
Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
"""
algebraic_ring = self.base_ring()
if isinstance(algebraic_ring, ExactReals):
algebraic_ring = algebraic_ring.base_ring()
if algebraic_ring not in [ZZ, QQ]:
algebraic_ring = algebraic_ring.number_field
return algebraic_ring
def decomposition(self, vector, base=None):
r"""
Write ``vector`` as a shortest sum `v = \sum a_i v_i` where the
`a_i` are coefficients from our ``base_ring()`` and the `v_i` are
vectors with entries in ``base``.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: V = Vectors(ZZ)
sage: V.decomposition([13, 37])
[(1, (13, 37))]
sage: V.decomposition([0, 0])
[]
::
sage: from pyexactreal import ExactReals
sage: R = ExactReals()
sage: V = Vectors(R)
sage: V.decomposition([R.random_element(), R.random_element()])
[(ℝ(0.178808…), (1, 0)), (ℝ(0.478968…), (0, 1))]
::
sage: from pyeantic import RealEmbeddedNumberField
sage: K = NumberField(x^3 - 2, 'a', embedding=AA(2)**(1/3))
sage: V = Vectors(K)
sage: V.decomposition([0, 1])
[(1, (0, 1))]
::
sage: R = ExactReals(K)
sage: V = Vectors(R)
sage: V.decomposition([K.random_element() * R.random_element(), R.random_element(), R.random_element()])
[(ℝ(0.621222…), (0, 1, 0)),
(ℝ(0.673083…), (0, 0, 1)),
(ℝ(0.782515…), (-1/2*a^2 - 4, 0, 0))]
"""
vector = tuple(self._to_coordinate(x) for x in vector)
if not any(vector): return []
if base is None:
base = self._algebraic_ring()
if base is self.base_ring():
return [(self.base_ring().one(), tuple(map(base, vector)))]
if hasattr(self.base_ring(), "number_field") and self.base_ring().number_field is base:
return [(self.base_ring().one(), tuple(base(self.base_ring()(x)) for x in vector))]
if isinstance(self.base_ring(), ExactReals):
from functools import reduce
span = type(vector[0].module()).span
module = reduce(lambda x,y: span(x, y.module()), vector, vector[0].module())
vector = [entry.promote(module).coefficients() for entry in vector]
vector = [[base(Vectors(base).base_ring()(c)) for c in coefficients] for coefficients in vector]
V = base**len(vector)
return [(self.base_ring()(module.gen(i)), tuple(V(coefficients))) for (i, coefficients) in enumerate(zip(*vector)) if any(coefficients)]
raise NotImplementedError("cannot decompose vector in %s over %s"%(self, base))
def _to_coordinate(self, x):
r"""
Convert ``x`` to something that the flatsurf backend for this vector type understands.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: V = Vectors(QQ)
sage: type(V._to_coordinate(1))
<class cppyy.gbl.__gmp_expr<__mpz_struct[1],__mpz_struct[1]> at ...>
sage: type(V._to_coordinate(1/2))
<class cppyy.gbl.__gmp_expr<__mpq_struct[1],__mpq_struct[1]> at ...>
"""
if isinstance(x, self.coordinate):
return x
if isinstance(x, cppyy.gbl.mpz_class):
return x
if isinstance(x, cppyy.gbl.mpq_class):
return x
if isinstance(self.base_ring(), real_embedded_number_field.RealEmbeddedNumberField):
# We create a copy of the coordinate. Otherwise, cppyy (as of
# 1.9.5) treats this as an r-value and moves x out even if it is
# referenced elsewhere.
value = self.base_ring()(x).renf_elem
return type(value)(value)
if isinstance(self.base_ring(), ExactReals):
return self.base_ring()(x)._backend
if x in ZZ:
return cppyy.gbl.mpz_class(str(x))
if x in QQ:
return cppyy.gbl.mpq_class(str(x))
raise NotImplementedError("Cannot convert %s to something the flatsurf backend understands yet, i.e., cannot convert a %s into a %s"%(x, type(x), type(self.coordinate)))
def _repr_(self):
r"""
Return a printable representation of this vector space.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: V = Vectors(QQ)
sage: V
Flatsurf Vectors over Rational Field
"""
return "Flatsurf Vectors over %r"%(self.base_ring(),)
def an_element(self):
return self((1,0))
def zero(self):
r"""
Return the zero vector.
EXAMPLES::
sage: from pyflatsurf.vector import Vectors
sage: V = Vectors(QQ)
sage: V.zero()
(0, 0)
"""
return self((0,0))
def is_reducible(n, den_tuple):
r"""
If (x1+x2+...+xd) is not present, use a linear relation to create it. Then
try to kill using other forms. Then try to write as P(x1,x2,...,x_{d-1}) (x1+x2+...+xd) and recurse.
Should solve all Tonrheim
EXAMPLES::
sage: from surface_dynamics.misc.generalized_multiple_zeta_values import linear_forms, is_reducible
sage: va, vb, vd, vc, ve, vf, vg = linear_forms(3)
sage: is_reducible(3, ((va,3),(vb,3),(vc,3)))
[(1, (((0, 0, 1), 3), ((0, 1, 1), 3), ((1, 1, 1), 3))),
(1, (((0, 1, 0), 3), ((0, 1, 1), 3), ((1, 1, 1), 3))),
(3, (((0, 0, 1), 2), ((0, 1, 1), 4), ((1, 1, 1), 3))),
(3, (((0, 1, 0), 2), ((0, 1, 1), 4), ((1, 1, 1), 3))),
...
(90, (((1, 0, 0), 1), ((1, 0, 1), 1), ((1, 1, 1), 7))),
(90, (((0, 1, 0), 1), ((1, 1, 0), 1), ((1, 1, 1), 7))),
(90, (((1, 0, 0), 1), ((1, 1, 0), 1), ((1, 1, 1), 7)))]
"""
F = FreeModule(ZZ, n)
vmax = F([1] * n)
vmax.set_immutable()
if len(den_tuple) == 1:
return
# force the max vector (1,1,...,1) to appear
imax = None
for i, (v, p) in enumerate(den_tuple):
if v == vmax:
imax = i
break
if imax is None:
imax = len(den_tuple)
den_tuple = den_tuple + ((vmax, 0), )
if imax != len(den_tuple) - 1:
den_tuple = list(den_tuple)
den_tuple.append(den_tuple.pop(imax))
den_tuple = tuple(den_tuple)
imax = len(den_tuple) - 1
assert den_tuple[imax][0] == vmax
M = matrix(QQ, [v for v, p in den_tuple if v != vmax])
try:
relation = M.solve_left(vmax)
except ValueError:
if den_tuple[-1][1] == 0:
return
den_tuple2 = den_tuple[:-1]
variables = set().union(*[[i for i in range(n) if v[i]]
for v, p in den_tuple2])
if len(variables) == n:
return
killed = [(1, den_tuple)]
else:
killed = kill_relation(n, den_tuple, imax, list(relation) + [0])
ans = defaultdict(QQ)
for coeff, den_tuple2 in killed:
assert den_tuple2[-1][0] == vmax
pmax = den_tuple2[-1][1]
assert pmax
den_tuple2 = den_tuple2[:-1]
variables = set().union(*[[i for i in range(n) if v[i]]
for v, p in den_tuple2])
if len(variables) == n:
# removing the maximal vector (1,1,...,1) is not enough to make a variable disappear
data = is_reducible(n, den_tuple2)
if data is None:
data = [(1, den_tuple2)]
else:
# less variables!
nn = len(variables)
variables = sorted(variables)
new_indices = {j: i for i, j in enumerate(variables)}
G = FreeModule(ZZ, nn)
new_den_tuple2 = []
for v, p in den_tuple2:
vv = G([v[i] for i in variables])
vv.set_immutable()
new_den_tuple2.append((vv, p))
new_den_tuple2 = tuple(new_den_tuple2)
data = is_reducible(nn, new_den_tuple2)
if data is None:
data = [(1, new_den_tuple2)]
# lift to the n variables version
new_data = []
for coeff3, den_tuple3 in data:
den_tuple3 = [(F([
v[new_indices[j]] if j in new_indices else 0
for j in range(n)
]), p) for v, p in den_tuple3]
for v, p in den_tuple3:
v.set_immutable()
new_data.append((coeff3, tuple(sorted(den_tuple3))))
data = new_data
# update the answer
for coeff3, den_tuple3 in data:
imax = None
for i, (v, p) in enumerate(den_tuple3):
if v == vmax:
imax = i
break
if imax is None:
den_tuple3 = den_tuple3 + ((vmax, pmax), )
else:
den_tuple3 = den_tuple3[:imax] + (
(vmax,
den_tuple3[imax][1] + pmax), ) + den_tuple3[imax + 1:]
ans[den_tuple3] += coeff * coeff3
if len(ans) > 1:
return [(coeff, den_tuple) for den_tuple, coeff in ans.items()]