def affine_subspace(V, a, m, ref=False):
n = V.degree()
diff = n - m.degree() - 1
base = [x**i * m for i in range(diff + 1)]
if ref:
base = [p(b) for b in base]
W = V.subspace([
V(list(base[i]) + [0] * (n - base[i].degree() - 1))
for i in range(diff + 1)
])
if ref:
a = p(a)
return AffineSubspace(V(a.list() + [0] * (n - a.degree() - 1)), W)
def _affine_subspace(self):
"""
Return the hyperplane as affine subspace.
OUTPUT:
The hyperplane as a
:class:`~sage.geometry.hyperplane_arrangement.affine_subspace.AffineSubspace`.
EXAMPLES::
sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: h = -1/3*x + 1/2*y - 1; h
Hyperplane -1/3*x + 1/2*y - 1
sage: h._affine_subspace()
Affine space p + W where:
p = (-12/13, 18/13)
W = Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[ 1 2/3]
"""
from sage.geometry.hyperplane_arrangement.affine_subspace import AffineSubspace
return AffineSubspace(self.point(), self.linear_part())
def _para_intersection_poset(A):
from sage.geometry.hyperplane_arrangement.affine_subspace import AffineSubspace
from sage.all import exists, flatten, Set, QQ, VectorSpace, Poset
from .Globals import __SANITY
N = _N
K = A.base_ring()
whole_space = AffineSubspace(0, VectorSpace(K, A.dimension()))
# L is the ranked list of affine subspaces in L(A).
L = [[whole_space], list(map(lambda H: H._affine_subspace(), A))]
# hyp_cont is the ranked list describing which hyperplanes contain the
# corresponding intersection.
hyp_cont = [[Set([])], [Set([k]) for k in range(len(A))]]
c = A.is_central() * (-1)
for r in range(2, A.rank() + c + 1):
print("{1}Working on elements of rank {0}".format(r, _time()))
m = len(L[r - 1])
pmax = lambda k: (k + 1) * (m // N) + (k == N - 1) * (m % N)
pmin = lambda k: k * (m // N)
all_input = lambda k: tuple([
A,
range(pmin(k), pmax(k)), hyp_cont[r - 1][pmin(k):pmax(k)], L[r - 1]
[pmin(k):pmax(k)]
])
data = list(
build_next([all_input(k) for k in range(N) if pmin(k) != pmax(k)]))
data = _reduce(lambda x, y: x + y[1], data, [])
new_lev, new_hyp = list(zip(*data))
new_lev = list(new_lev)
new_hyp = list(new_hyp)
i = 0
# Merge the lists down
print("{0}Merging the lists from the {1} workers".format(_time(), N))
# First we check the affine spaces
while i < len(new_lev):
U = new_lev[i]
B1 = U.linear_part().basis_matrix()
p1 = U.point()
j = i + 1
while j < len(new_lev):
V = new_lev[j]
B2 = V.linear_part().basis_matrix()
p2 = V.point()
if B1 == B2 and p1 == p2:
new_lev = new_lev[:j] + new_lev[j + 1:]
new_hyp[i] = new_hyp[i].union(new_hyp[j])
new_hyp = new_hyp[:j] + new_hyp[j + 1:]
else:
j += 1
i += 1
# Second we check the labels of intersection (don't want duplicates)
i = 0
while i < len(new_lev):
j = i + 1
while j < len(new_lev):
if new_hyp[i] == new_hyp[j]:
new_lev = new_lev[:j] + new_lev[j + 1:]
new_hyp = new_hyp[:j] + new_hyp[j + 1:]
else:
j += 1
i += 1
L.append(new_lev)
hyp_cont.append(new_hyp)
# A silly optimization for centrals.
if A.is_central() and len(A) > 1:
inter = lambda X, Y: X.intersection(Y._affine_subspace())
L.append([_reduce(inter, A[1:], A[0]._affine_subspace())])
hyp_cont.append([Set(list(range(len(A))))])
L_flat = list(_reduce(lambda x, y: x + y, L, []))
hc_flat = list(_reduce(lambda x, y: x + y, hyp_cont, []))
# Sanity checks
if __SANITY:
print("{0}Running sanity check".format(_time()))
assert len(L_flat) == len(hc_flat)
for i in range(len(hc_flat)):
for j in range(i + 1, len(hc_flat)):
assert hc_flat[i] != hc_flat[j], "{0} vs {1}".format(i, j)
for i in range(len(L_flat)):
I = list(map(lambda x: A[x], hc_flat[i]))
U = _reduce(lambda x, y: x.intersection(y._affine_subspace()), I,
whole_space)
assert U == L_flat[i], "{0} vs {1}".format(U, L_flat[i])
print("{0}Constructing lattice of flats".format(_time()))
t = {}
for i in range(len(hc_flat)):
t[i] = Set(list(map(lambda x: x + 1, hc_flat[i])))
cmp_fn = lambda p, q: t[p].issubset(t[q])
label_dict = {i: t[i] for i in range(len(hc_flat))}
get_hyp = lambda i: A[label_dict[i].an_element() - 1]
hyp_dict = {i + 1: get_hyp(i + 1) for i in range(len(A))}
return [Poset((t, cmp_fn)), label_dict, hyp_dict]