def _check_generic(self, cones1, cones2, v):
"""
Returns pairs of cones which intersect with respect to `v`.
For every pair of cones `c1`, `c2` from respective lists, check whether
they intersect generically with respect to `v`.
Return list of ordered pairs of cones which intersect with respect to `v`.
This means the intersection of `c1` and `c2+v` is non-empty.
EXAMPLES::
sage: A = MW(toric_varieties.P2().fan())
sage: cones = A._fan(1)
sage: v = (.1,.3)
sage: A._check_generic(cones,cones,v)
[(1-d cone of Rational polyhedral fan in 2-d lattice N, 1-d cone of Rational polyhedral fan in 2-d lattice N)]
"""
good_pairs = []
for c1 in cones1:
for c2 in cones2:
# check whether v is in c1 - c2 (i.e. mink sum of c1 and -c2)
C = Cone(rays=[r for r in c1.rays()] +
[-1 * r for r in c2.rays()])
if C.contains(v):
good_pairs.append((c1, c2))
return good_pairs
def cone_in_homology(tri, angle):
rays = taut_rays(tri, angle)
if len(rays) == 0:
return []
else:
A = projection_to_homology(tri,angle)
projectedRays = [A*v for v in rays]
C = Cone(projectedRays)
return [ray for ray in C.rays()]
def experiment():
rank = 5
level = 4
num_points = 11
client = cbc.CBClient()
liealg = cbd.TypeALieAlgebra(rank, store_fusion=True, exact=True)
rays = []
wts = []
ranks = []
for wt in liealg.get_weights(level):
if wt == tuple([0 for x in range(0, rank)]):
continue
cbb = cbd.SymmetricConformalBlocksBundle(client, liealg, wt,
num_points, level)
if cbb.get_rank() == 0:
continue
divisor = cbb.get_symmetrized_divisor()
if divisor == [
sage.Rational(0) for x in range(0, num_points // 2 - 1)
]:
continue
rays = rays + [divisor]
wts = wts + [wt]
ranks = ranks + [cbb.get_rank()]
#print(wt, cbb.get_rank(), divisor)
p = Polyhedron(rays=rays, base_ring=RationalField(), backend="cdd")
extremal_rays = [list(v.vector()) for v in p.Vrepresentation()]
c = Cone(p)
print("Extremal rays:")
for i in range(0, len(rays)):
ray = rays[i]
wt = wts[i]
rk = ranks[i]
if ray in extremal_rays:
print(rk, wt, ray)
def fan_isomorphism_generator(fan1, fan2):
"""
Iterate over the isomorphisms from ``fan1`` to ``fan2``.
ALGORITHM:
The :meth:`sage.geometry.fan.Fan.vertex_graph` of the two fans is
compared. For each graph isomorphism, we attempt to lift it to an
actual isomorphism of fans.
INPUT:
- ``fan1``, ``fan2`` -- two fans.
OUTPUT:
Yields the fan isomorphisms as matrices acting from the right on
rays.
EXAMPLES::
sage: fan = toric_varieties.P2().fan()
sage: from sage.geometry.fan_isomorphism import fan_isomorphism_generator
sage: tuple( fan_isomorphism_generator(fan, fan) )
(
[1 0] [0 1] [ 1 0] [ 0 1] [-1 -1] [-1 -1]
[0 1], [1 0], [-1 -1], [-1 -1], [ 1 0], [ 0 1]
)
sage: m1 = matrix([(1, 0), (0, -5), (-3, 4)])
sage: m2 = matrix([(3, 0), (1, 0), (-2, 1)])
sage: m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0]
True
sage: fan1 = Fan([Cone([m1*vector([23, 14]), m1*vector([ 3,100])]),
... Cone([m1*vector([-1,-14]), m1*vector([-100, -5])])])
sage: fan2 = Fan([Cone([m2*vector([23, 14]), m2*vector([ 3,100])]),
... Cone([m2*vector([-1,-14]), m2*vector([-100, -5])])])
sage: next(fan_isomorphism_generator(fan1, fan2))
[18 1 -5]
[ 4 0 -1]
[ 5 0 -1]
sage: m0 = identity_matrix(ZZ, 2)
sage: m1 = matrix([(1, 0), (0, -5), (-3, 4)])
sage: m2 = matrix([(3, 0), (1, 0), (-2, 1)])
sage: m1.elementary_divisors() == m2.elementary_divisors() == [1,1,0]
True
sage: fan0 = Fan([Cone([m0*vector([1,0]), m0*vector([1,1])]),
... Cone([m0*vector([1,1]), m0*vector([0,1])])])
sage: fan1 = Fan([Cone([m1*vector([1,0]), m1*vector([1,1])]),
... Cone([m1*vector([1,1]), m1*vector([0,1])])])
sage: fan2 = Fan([Cone([m2*vector([1,0]), m2*vector([1,1])]),
... Cone([m2*vector([1,1]), m2*vector([0,1])])])
sage: tuple(fan_isomorphism_generator(fan0, fan0))
(
[1 0] [0 1]
[0 1], [1 0]
)
sage: tuple(fan_isomorphism_generator(fan1, fan1))
(
[1 0 0] [ -3 -20 28]
[0 1 0] [ -1 -4 7]
[0 0 1], [ -1 -5 8]
)
sage: tuple(fan_isomorphism_generator(fan1, fan2))
(
[18 1 -5] [ 6 -3 7]
[ 4 0 -1] [ 1 -1 2]
[ 5 0 -1], [ 2 -1 2]
)
sage: tuple(fan_isomorphism_generator(fan2, fan1))
(
[ 0 -1 1] [ 0 -1 1]
[ 1 -7 2] [ 2 -2 -5]
[ 0 -5 4], [ 1 0 -3]
)
"""
if not fan_isomorphic_necessary_conditions(fan1, fan2):
return
graph1 = fan1.vertex_graph()
graph2 = fan2.vertex_graph()
graph_iso = graph1.is_isomorphic(graph2, edge_labels=True, certify=True)
if not graph_iso[0]:
return
graph_iso = graph_iso[1]
# Pick a basis of rays in fan1
max_cone = fan1(fan1.dim())[0]
fan1_pivot_rays = max_cone.rays()
fan1_basis = fan1_pivot_rays + fan1.virtual_rays() # A QQ-basis for N_1
fan1_pivot_cones = [fan1.embed(Cone([r])) for r in fan1_pivot_rays]
# The fan2 cones as set(set(ray indices))
fan2_cones = frozenset(
frozenset(cone.ambient_ray_indices())
for cone in fan2.generating_cones())
# iterate over all graph isomorphisms graph1 -> graph2
for perm in graph2.automorphism_group(edge_labels=True):
# find a candidate m that maps fan1_basis to the image rays under the graph isomorphism
fan2_pivot_cones = [perm(graph_iso[c]) for c in fan1_pivot_cones]
fan2_pivot_rays = fan2.rays(
[c.ambient_ray_indices()[0] for c in fan2_pivot_cones])
fan2_basis = fan2_pivot_rays + fan2.virtual_rays()
try:
m = matrix(ZZ, fan1_basis).solve_right(matrix(ZZ, fan2_basis))
m = m.change_ring(ZZ)
except (ValueError, TypeError):
continue # no solution
# check that the candidate m lifts the vertex graph homomorphism
graph_image_ray_indices = [
perm(graph_iso[c]).ambient_ray_indices()[0] for c in fan1(1)
]
try:
matrix_image_ray_indices = [
fan2.rays().index(r * m) for r in fan1.rays()
]
except ValueError:
continue
if graph_image_ray_indices != matrix_image_ray_indices:
continue
# check that the candidate m maps generating cone to generating cone
image_cones = frozenset( # The image(fan1) cones as set(set(integers)
frozenset(graph_image_ray_indices[i]
for i in cone.ambient_ray_indices())
for cone in fan1.generating_cones())
if image_cones == fan2_cones:
m.set_immutable()
yield m
def borcherds_input_Qbasis(self, pole_order, prec, verbose=False):
r"""
Compute a Q-basis of input functions into the Borcherds lift with pole order in infinity up to pole_order.
This method computes a list of Borcherds lift inputs F_0, ..., F_d which is a Q-basis in the following sense: it is minimal with the property that every modular form with pole order at most pole_order whose Borcherds lift is holomorphic can be expressed in the form (k_0 F_0 + ... + k_d F_d) where k_i are nonnegative rational numbers.
INPUT:
- ``pole_order`` -- positive number (does not need to be an integer)
- ``prec`` -- precision of the output
OUTPUT: WeilRepModularFormsBasis
"""
S = self.gram_matrix()
w = self.weilrep()
K = w.base_field()
d = K.discriminant()
wt = self.input_wt()
M, p, X = self._borcherds_product_polyhedron(pole_order,
prec,
verbose=verbose)
try:
b = Matrix(Cone(p).rays())
if verbose:
print('I will now try to find Borcherds product inputs.')
try:
u = M.solve_left(b)
Y = [v * X for v in u.rows()]
Y = WeilRepModularFormsBasis(wt, Y, w)
except ValueError:
Y = WeilRepModularFormsBasis(wt, [], w)
except IndexError:
Y = WeilRepModularFormsBasis(wt, [], w)
if wt >= 0:
X = deepcopy(
w.basis_vanishing_to_order(wt, max(0, -pole_order), prec))
if X:
X.extend(Y)
Y = X
X = Y._WeilRepModularFormsBasis__basis
if d >= -4:
if d == -4:
f = w.multiplication_by_i()
n = 2
else:
f = w.multiplication_by_zeta()
f2 = f * f
n = 3
for i, x in enumerate(X):
try:
j = X.index(f(x))
if j == i:
X[i] = x / n
else:
del X[j]
except ValueError:
pass
if n == 3:
try:
j2 = X.index(f2(x))
del X[j2]
except ValueError:
pass
X.sort(key=lambda x: x.fourier_expansion()[0][2][0])
return WeilRepModularFormsBasis(wt, X, w)
def trivial(ambient_dim=None, lattice=None):
r"""
The trivial cone with no nonzero generators in ``ambient_dim``
dimensions, or living in ``lattice``.
INPUT:
- ``ambient_dim`` -- a nonnegative integer (default: ``None``); the
dimension of the ambient space
- ``lattice`` -- a toric lattice (default: ``None``); the lattice in
which the cone will live
If ``ambient_dim`` is omitted, then it will be inferred from the
rank of ``lattice``. If the ``lattice`` is omitted, then the
default lattice of rank ``ambient_dim`` will be used.
A ``ValueError`` is raised if neither ``ambient_dim`` nor
``lattice`` are specified. It is also a ``ValueError`` to specify
both ``ambient_dim`` and ``lattice`` unless the rank of
``lattice`` is equal to ``ambient_dim``.
OUTPUT:
A :class:`.ConvexRationalPolyhedralCone` representing the
trivial cone with no nonzero generators living in ``lattice``,
with ambient dimension ``ambient_dim``.
A ``ValueError`` can be raised if the inputs are incompatible or
insufficient. See the INPUT documentation for details.
EXAMPLES:
Construct the trivial cone, containing only the origin, in three
dimensions::
sage: cones.trivial(3)
0-d cone in 3-d lattice N
If a ``lattice`` is given, the trivial cone will live in that
lattice::
sage: L = ToricLattice(3, 'M')
sage: cones.trivial(3, lattice=L)
0-d cone in 3-d lattice M
TESTS:
We can construct the trivial cone in a trivial ambient space::
sage: cones.trivial(0)
0-d cone in 0-d lattice N
An error is raised if the rank of the lattice disagrees with
``ambient_dim``::
sage: L = ToricLattice(1, 'M')
sage: cones.trivial(3, lattice=L)
Traceback (most recent call last):
...
ValueError: lattice rank=1 and ambient_dim=3 are incompatible
We also get an error if no arguments are given::
sage: cones.trivial()
Traceback (most recent call last):
...
ValueError: either the ambient dimension or the lattice must
be specified
"""
from sage.geometry.cone import Cone
(ambient_dim, lattice) = _preprocess_args(ambient_dim, lattice)
return Cone([], lattice)
def schur(ambient_dim=None, lattice=None):
r"""
The Schur cone in ``ambient_dim`` dimensions, or living
in ``lattice``.
The Schur cone in `n` dimensions induces the majorization ordering
on the ambient space. If `\left\{e_{1}, e_{2}, \ldots,
e_{n}\right\}` is the standard basis for the space, then its
generators are `\left\{e_{i} - e_{i+1}\ |\ 1 \le i \le
n-1\right\}`. Its dual is the downward monotonic cone.
INPUT:
- ``ambient_dim`` -- a nonnegative integer (default: ``None``); the
dimension of the ambient space
- ``lattice`` -- a toric lattice (default: ``None``); the lattice in
which the cone will live
If ``ambient_dim`` is omitted, then it will be inferred from the
rank of ``lattice``. If the ``lattice`` is omitted, then the
default lattice of rank ``ambient_dim`` will be used.
A ``ValueError`` is raised if neither ``ambient_dim`` nor
``lattice`` are specified. It is also a ``ValueError`` to specify
both ``ambient_dim`` and ``lattice`` unless the rank of
``lattice`` is equal to ``ambient_dim``.
OUTPUT:
A :class:`.ConvexRationalPolyhedralCone` representing the Schur
cone living in ``lattice``, with ambient dimension ``ambient_dim``.
Each generating ray has the integer ring as its base ring.
A ``ValueError`` can be raised if the inputs are incompatible or
insufficient. See the INPUT documentation for details.
REFERENCES:
- [GS2010]_, Section 3.1
- [IS2005]_, Example 7.3
- [SS2016]_, Example 7.4
EXAMPLES:
Verify the claim [SS2016]_ that the maximal angle between any two
generators of the Schur cone and the nonnegative orthant in
dimension five is `\left(3/4\right)\pi`::
sage: P = cones.schur(5)
sage: Q = cones.nonnegative_orthant(5)
sage: G = ( g.change_ring(QQbar).normalized() for g in P )
sage: H = ( h.change_ring(QQbar).normalized() for h in Q )
sage: actual = max(arccos(u.inner_product(v)) for u in G for v in H)
sage: expected = 3*pi/4
sage: abs(actual - expected).n() < 1e-12
True
The dual of the Schur cone is the "downward monotonic cone"
[GS2010]_, whose elements' entries are in non-increasing order::
sage: set_random_seed()
sage: ambient_dim = ZZ.random_element(10)
sage: K = cones.schur(ambient_dim).dual()
sage: x = K.random_element()
sage: all( x[i] >= x[i+1] for i in range(ambient_dim-1) )
True
TESTS:
We get the trivial cone when ``ambient_dim`` is zero::
sage: cones.schur(0).is_trivial()
True
The Schur cone induces the majorization ordering, as in Iusem
and Seeger's [IS2005]_ Example 7.3::
sage: set_random_seed()
sage: def majorized_by(x,y):
....: return (all(sum(x[0:i]) <= sum(y[0:i])
....: for i in range(x.degree()-1))
....: and sum(x) == sum(y))
sage: ambient_dim = ZZ.random_element(10)
sage: V = VectorSpace(QQ, ambient_dim)
sage: S = cones.schur(ambient_dim)
sage: majorized_by(V.zero(), S.random_element())
True
sage: x = V.random_element()
sage: y = V.random_element()
sage: majorized_by(x,y) == ( (y-x) in S )
True
If a ``lattice`` was given, it is actually used::
sage: L = ToricLattice(3, 'M')
sage: cones.schur(3, lattice=L)
2-d cone in 3-d lattice M
Unless the rank of the lattice disagrees with ``ambient_dim``::
sage: L = ToricLattice(1, 'M')
sage: cones.schur(3, lattice=L)
Traceback (most recent call last):
...
ValueError: lattice rank=1 and ambient_dim=3 are incompatible
We also get an error if no arguments are given::
sage: cones.schur()
Traceback (most recent call last):
...
ValueError: either the ambient dimension or the lattice must
be specified
"""
from sage.geometry.cone import Cone
from sage.matrix.constructor import matrix
from sage.rings.integer_ring import ZZ
(ambient_dim, lattice) = _preprocess_args(ambient_dim, lattice)
def _f(i, j):
if i == j:
return 1
elif j - i == 1:
return -1
else:
return 0
# The "max" below catches the trivial case where ambient_dim == 0.
S = matrix(ZZ, max(0, ambient_dim - 1), ambient_dim, _f)
return Cone(S.rows(), lattice)
def rearrangement(p, ambient_dim=None, lattice=None):
r"""
The rearrangement cone of order ``p`` in ``ambient_dim``
dimensions, or living in ``lattice``.
The rearrangement cone of order ``p`` in ``ambient_dim``
dimensions consists of all vectors of length ``ambient_dim``
whose smallest ``p`` components sum to a nonnegative number.
For example, the rearrangement cone of order one has its single
smallest component nonnegative. This implies that all components
are nonnegative, and that therefore the rearrangement cone of
order one is the nonnegative orthant in its ambient space.
When ``p`` and ``ambient_dim`` are equal, all components of the
cone's elements must sum to a nonnegative number. In other
words, the rearrangement cone of order ``ambient_dim`` is a
half-space.
INPUT:
- ``p`` -- a nonnegative integer; the number of components to
"rearrange", between ``1`` and ``ambient_dim`` inclusive
- ``ambient_dim`` -- a nonnegative integer (default: ``None``); the
dimension of the ambient space
- ``lattice`` -- a toric lattice (default: ``None``); the lattice in
which the cone will live
If ``ambient_dim`` is omitted, then it will be inferred from the
rank of ``lattice``. If the ``lattice`` is omitted, then the
default lattice of rank ``ambient_dim`` will be used.
A ``ValueError`` is raised if neither ``ambient_dim`` nor
``lattice`` are specified. It is also a ``ValueError`` to specify
both ``ambient_dim`` and ``lattice`` unless the rank of
``lattice`` is equal to ``ambient_dim``.
It is also a ``ValueError`` to specify a non-integer ``p``.
OUTPUT:
A :class:`.ConvexRationalPolyhedralCone` representing the
rearrangement cone of order ``p`` living in ``lattice``, with
ambient dimension ``ambient_dim``. Each generating ray has the
integer ring as its base ring.
A ``ValueError`` can be raised if the inputs are incompatible or
insufficient. See the INPUT documentation for details.
ALGORITHM:
Suppose that the ambient space is of dimension `n`. The extreme
directions of the rearrangement cone for `1 \le p \le n-1` are
given by [Jeong2017]_ Theorem 5.2.3. When `2 \le p \le n-2` (that
is, if we ignore `p = 1` and `p = n-1`), they consist of
- the standard basis `\left\{e_{1},e_{2},\ldots,e_{n}\right\}` for
the ambient space, and
- the `n` vectors `\left(1,1,\ldots,1\right)^{T} - pe_{i}` for
`i = 1,2,\ldots,n`.
Special cases are then given for `p = 1` and `p = n-1` in the
theorem. However in SageMath we don't need conically-independent
extreme directions. We only need a generating set, because the
:func:`Cone` function will eliminate any redundant generators. And
one can easily verify that the special-case extreme directions for
`p = 1` and `p = n-1` are contained in the conic hull of the `2n`
generators just described. The half space resulting from `p = n`
is also covered by this set of generators, so for all valid `p` we
simply take the conic hull of those `2n` vectors.
REFERENCES:
- [GJ2016]_, Section 4
- [HS2010]_, Example 2.21
- [Jeong2017]_, Section 5.2
EXAMPLES:
The rearrangement cones of order one are nonnegative orthants::
sage: orthant = cones.nonnegative_orthant(6)
sage: cones.rearrangement(1,6).is_equivalent(orthant)
True
When ``p`` and ``ambient_dim`` are equal, the rearrangement cone
is a half-space, so we expect its lineality to be one less than
``ambient_dim`` because it will contain a hyperplane but is not
the entire space::
sage: cones.rearrangement(5,5).lineality()
4
Jeong's Proposition 5.2.1 [Jeong2017]_ states that all rearrangement
cones are proper when ``p`` is less than ``ambient_dim``::
sage: all( cones.rearrangement(p, ambient_dim).is_proper()
....: for ambient_dim in range(10)
....: for p in range(1, ambient_dim) )
True
Jeong's Corollary 5.2.4 [Jeong2017]_ states that if `p = n-1` in
an `n`-dimensional ambient space, then the Lyapunov rank of the
rearrangement cone is `n`, and that for all other `p > 1` its
Lyapunov rank is one::
sage: all( cones.rearrangement(p, ambient_dim).lyapunov_rank()
....: ==
....: ambient_dim
....: for ambient_dim in range(2, 10)
....: for p in [ ambient_dim-1 ] )
True
sage: all( cones.rearrangement(p, ambient_dim).lyapunov_rank() == 1
....: for ambient_dim in range(3, 10)
....: for p in range(2, ambient_dim-1) )
True
TESTS:
Jeong's Proposition 5.2.1 [Jeong2017]_ states that rearrangement
cones are permutation-invariant::
sage: ambient_dim = ZZ.random_element(2,10).abs()
sage: p = ZZ.random_element(1, ambient_dim)
sage: K = cones.rearrangement(p, ambient_dim)
sage: P = SymmetricGroup(ambient_dim).random_element().matrix()
sage: all( K.contains(P*r) for r in K )
True
The smallest ``p`` components of every element of the rearrangement
cone should sum to a nonnegative number. In other words, the
generators really are what we think they are::
sage: set_random_seed()
sage: def _has_rearrangement_property(v,p):
....: return sum( sorted(v)[0:p] ) >= 0
sage: all(
....: _has_rearrangement_property(
....: cones.rearrangement(p, ambient_dim).random_element(),
....: p
....: )
....: for ambient_dim in range(2, 10)
....: for p in range(1, ambient_dim+1)
....: )
True
The rearrangement cone of order ``p`` is, almost by definition,
contained in the rearrangement cone of order ``p + 1``::
sage: set_random_seed()
sage: ambient_dim = ZZ.random_element(2,10)
sage: p = ZZ.random_element(1, ambient_dim)
sage: K1 = cones.rearrangement(p, ambient_dim)
sage: K2 = cones.rearrangement(p+1, ambient_dim)
sage: all( x in K2 for x in K1 )
True
Jeong's Proposition 5.2.1 [Jeong2017]_ states that the rearrangement
cone of order ``p`` is linearly isomorphic to the rearrangement
cone of order ``ambient_dim - p`` when ``p`` is less than
``ambient_dim``::
sage: set_random_seed()
sage: ambient_dim = ZZ.random_element(2,10)
sage: p = ZZ.random_element(1, ambient_dim)
sage: K1 = cones.rearrangement(p, ambient_dim)
sage: K2 = cones.rearrangement(ambient_dim-p, ambient_dim)
sage: Mp = ((1/p)*matrix.ones(QQ, ambient_dim)
....: - matrix.identity(QQ, ambient_dim))
sage: Cone( (Mp*K2.rays()).columns() ).is_equivalent(K1)
True
The order ``p`` should be an integer between ``1`` and
``ambient_dim``, inclusive::
sage: cones.rearrangement(0,3)
Traceback (most recent call last):
...
ValueError: order p=0 should be an integer between 1 and
ambient_dim=3, inclusive
sage: cones.rearrangement(5,3)
Traceback (most recent call last):
...
ValueError: order p=5 should be an integer between 1 and
ambient_dim=3, inclusive
sage: cones.rearrangement(3/2, 3)
Traceback (most recent call last):
...
ValueError: order p=3/2 should be an integer between 1 and
ambient_dim=3, inclusive
If a ``lattice`` was given, it is actually used::
sage: L = ToricLattice(3, 'M')
sage: cones.rearrangement(2, 3, lattice=L)
3-d cone in 3-d lattice M
Unless the rank of the lattice disagrees with ``ambient_dim``::
sage: L = ToricLattice(1, 'M')
sage: cones.rearrangement(2, 3, lattice=L)
Traceback (most recent call last):
...
ValueError: lattice rank=1 and ambient_dim=3 are incompatible
We also get an error if neither the ambient dimension nor lattice
are specified::
sage: cones.rearrangement(3)
Traceback (most recent call last):
...
ValueError: either the ambient dimension or the lattice must
be specified
"""
from sage.geometry.cone import Cone
from sage.matrix.constructor import matrix
from sage.rings.integer_ring import ZZ
(ambient_dim, lattice) = _preprocess_args(ambient_dim, lattice)
if p < 1 or p > ambient_dim or p not in ZZ:
raise ValueError("order p=%s should be an integer between 1 "
"and ambient_dim=%d, inclusive" % (p, ambient_dim))
I = matrix.identity(ZZ, ambient_dim)
M = matrix.ones(ZZ, ambient_dim) - p * I
G = matrix.identity(ZZ, ambient_dim).rows() + M.rows()
return Cone(G, lattice=lattice)
def nonnegative_orthant(ambient_dim=None, lattice=None):
r"""
The nonnegative orthant in ``ambient_dim`` dimensions, or living
in ``lattice``.
The nonnegative orthant consists of all componentwise-nonnegative
vectors. It is the convex-conic hull of the standard basis.
INPUT:
- ``ambient_dim`` -- a nonnegative integer (default: ``None``); the
dimension of the ambient space
- ``lattice`` -- a toric lattice (default: ``None``); the lattice in
which the cone will live
If ``ambient_dim`` is omitted, then it will be inferred from the
rank of ``lattice``. If the ``lattice`` is omitted, then the
default lattice of rank ``ambient_dim`` will be used.
A ``ValueError`` is raised if neither ``ambient_dim`` nor
``lattice`` are specified. It is also a ``ValueError`` to specify
both ``ambient_dim`` and ``lattice`` unless the rank of
``lattice`` is equal to ``ambient_dim``.
OUTPUT:
A :class:`.ConvexRationalPolyhedralCone` living in ``lattice``
and having ``ambient_dim`` standard basis vectors as its
generators. Each generating ray has the integer ring as its
base ring.
A ``ValueError`` can be raised if the inputs are incompatible or
insufficient. See the INPUT documentation for details.
REFERENCES:
- Chapter 2 in [BV2009]_ (Examples 2.4, 2.14, and 2.23 in particular)
EXAMPLES::
sage: cones.nonnegative_orthant(3).rays()
N(1, 0, 0),
N(0, 1, 0),
N(0, 0, 1)
in 3-d lattice N
TESTS:
We can construct the trivial cone as the nonnegative orthant in a
trivial vector space::
sage: cones.nonnegative_orthant(0)
0-d cone in 0-d lattice N
The nonnegative orthant is a proper cone::
sage: set_random_seed()
sage: ambient_dim = ZZ.random_element(10)
sage: K = cones.nonnegative_orthant(ambient_dim)
sage: K.is_proper()
True
If a ``lattice`` was given, it is actually used::
sage: L = ToricLattice(3, 'M')
sage: cones.nonnegative_orthant(lattice=L)
3-d cone in 3-d lattice M
Unless the rank of the lattice disagrees with ``ambient_dim``::
sage: L = ToricLattice(1, 'M')
sage: cones.nonnegative_orthant(3, lattice=L)
Traceback (most recent call last):
...
ValueError: lattice rank=1 and ambient_dim=3 are incompatible
We also get an error if no arguments are given::
sage: cones.nonnegative_orthant()
Traceback (most recent call last):
...
ValueError: either the ambient dimension or the lattice must
be specified
"""
from sage.geometry.cone import Cone
from sage.matrix.constructor import matrix
from sage.rings.integer_ring import ZZ
(ambient_dim, lattice) = _preprocess_args(ambient_dim, lattice)
I = matrix.identity(ZZ, ambient_dim)
return Cone(I.rows(), lattice)
def borcherds_input_Qbasis(self, pole_order, prec, verbose=False):
r"""
Compute a Q-basis of input functions into the Borcherds lift with pole order in infinity up to pole_order.
This method computes a list of Borcherds lift inputs F_0, ..., F_d which is a Q-basis in the following sense: it is minimal with the property that every modular form with pole order at most pole_order whose Borcherds lift is holomorphic can be expressed in the form (k_0 F_0 + ... + k_d F_d) where k_i are nonnegative rational numbers.
INPUT:
- ``pole_order`` -- positive number (does not need to be an integer)
- ``prec`` -- precision of the output
OUTPUT: WeilRepModularFormsBasis
EXAMPLES::
sage: from weilrep import *
sage: m = ParamodularForms(5)
sage: m.borcherds_input_Qbasis(1/4, 5)
[(0), 10 + 50*q + 260*q^2 + 1030*q^3 + 3500*q^4 + O(q^5)]
[(1/10), 6*q^(-1/20) + 16*q^(19/20) + 82*q^(39/20) + 304*q^(59/20) + 1046*q^(79/20) + 3120*q^(99/20) + O(q^(119/20))]
[(1/5), q^(-1/5) - 24*q^(4/5) - 143*q^(9/5) - 622*q^(14/5) - 2204*q^(19/5) - 6876*q^(24/5) + O(q^(29/5))]
[(3/10), -16*q^(11/20) - 102*q^(31/20) - 448*q^(51/20) - 1650*q^(71/20) - 5248*q^(91/20) + O(q^(111/20))]
[(2/5), -q^(1/5) + 14*q^(6/5) + 92*q^(11/5) + 386*q^(16/5) + 1318*q^(21/5) + O(q^(26/5))]
[(1/2), 20*q^(3/4) + 160*q^(7/4) + 700*q^(11/4) + 2560*q^(15/4) + 8000*q^(19/4) + O(q^(23/4))]
[(3/5), -q^(1/5) + 14*q^(6/5) + 92*q^(11/5) + 386*q^(16/5) + 1318*q^(21/5) + O(q^(26/5))]
[(7/10), -16*q^(11/20) - 102*q^(31/20) - 448*q^(51/20) - 1650*q^(71/20) - 5248*q^(91/20) + O(q^(111/20))]
[(4/5), q^(-1/5) - 24*q^(4/5) - 143*q^(9/5) - 622*q^(14/5) - 2204*q^(19/5) - 6876*q^(24/5) + O(q^(29/5))]
[(9/10), 6*q^(-1/20) + 16*q^(19/20) + 82*q^(39/20) + 304*q^(59/20) + 1046*q^(79/20) + 3120*q^(99/20) + O(q^(119/20))]
------------------------------------------------------------
[(0), 36 + 586*q + 3708*q^2 + 17694*q^3 + 68852*q^4 + O(q^5)]
[(1/10), -5*q^(-1/20) - 270*q^(19/20) - 1935*q^(39/20) - 10275*q^(59/20) - 43245*q^(79/20) - 157545*q^(99/20) + O(q^(119/20))]
[(1/5), 5*q^(-1/5) + 90*q^(4/5) + 1035*q^(9/5) + 6130*q^(14/5) + 28460*q^(19/5) + 109260*q^(24/5) + O(q^(29/5))]
[(3/10), -185*q^(11/20) - 1595*q^(31/20) - 8505*q^(51/20) - 36145*q^(71/20) - 131485*q^(91/20) + O(q^(111/20))]
[(2/5), 65*q^(1/5) + 630*q^(6/5) + 4100*q^(11/5) + 18940*q^(16/5) + 74070*q^(21/5) + O(q^(26/5))]
[(1/2), 7*q^(-1/4) - 54*q^(3/4) - 747*q^(7/4) - 5026*q^(11/4) - 23859*q^(15/4) - 94960*q^(19/4) + O(q^(23/4))]
[(3/5), 65*q^(1/5) + 630*q^(6/5) + 4100*q^(11/5) + 18940*q^(16/5) + 74070*q^(21/5) + O(q^(26/5))]
[(7/10), -185*q^(11/20) - 1595*q^(31/20) - 8505*q^(51/20) - 36145*q^(71/20) - 131485*q^(91/20) + O(q^(111/20))]
[(4/5), 5*q^(-1/5) + 90*q^(4/5) + 1035*q^(9/5) + 6130*q^(14/5) + 28460*q^(19/5) + 109260*q^(24/5) + O(q^(29/5))]
[(9/10), -5*q^(-1/20) - 270*q^(19/20) - 1935*q^(39/20) - 10275*q^(59/20) - 43245*q^(79/20) - 157545*q^(99/20) + O(q^(119/20))]
"""
S = self.gram_matrix()
w = self.weilrep()
wt = self.input_wt()
M, p, X = self._borcherds_product_polyhedron(pole_order,
prec,
verbose=verbose)
try:
b = matrix(Cone(p).rays())
if verbose:
print('I will now try to find Borcherds product inputs.')
try:
u = M.solve_left(b)
Y = [v * X for v in u.rows()]
Y.sort(key=lambda x: x.fourier_expansion()[0][2][0])
Y = WeilRepModularFormsBasis(wt, Y, w)
except ValueError:
Y = WeilRepModularFormsBasis(wt, [], w)
except IndexError:
Y = WeilRepModularFormsBasis(wt, [], w)
if wt >= 0:
X = X = deepcopy(
w.basis_vanishing_to_order(wt, max(0, -pole_order), prec))
if X:
X.extend(Y)
return X
return Y
def borcherds_input_basis(self, pole_order, prec, verbose=False):
r"""
Compute a basis of input functions into the Borcherds lift with pole up to pole_order.
This method computes a list of Borcherds lift inputs F_0, ..., F_d which is a basis in the following sense: it is minimal with the property that every modular form with pole order at most pole_order whose Borcherds lift is holomorphic can be expressed in the form (k_0 F_0 + ... + k_d F_d) where k_i are nonnegative integers.
WARNING: this can take a long time, and the output list might be longer than you expect!!
INPUT:
- ``pole_order`` -- positive number (does not need to be an integer)
- ``prec`` -- precision of the output
OUTPUT: WeilRepModularFormsBasis
EXAMPLES::
sage: from weilrep import *
sage: m = ParamodularForms(5)
sage: m.borcherds_input_basis(1/4, 5)
[(0), 8 + 98*q + 604*q^2 + 2822*q^3 + 10836*q^4 + O(q^5)]
[(1/10), q^(-1/20) - 34*q^(19/20) - 253*q^(39/20) - 1381*q^(59/20) - 5879*q^(79/20) - 21615*q^(99/20) + O(q^(119/20))]
[(1/5), q^(-1/5) + 6*q^(4/5) + 107*q^(9/5) + 698*q^(14/5) + 3436*q^(19/5) + 13644*q^(24/5) + O(q^(29/5))]
[(3/10), -31*q^(11/20) - 257*q^(31/20) - 1343*q^(51/20) - 5635*q^(71/20) - 20283*q^(91/20) + O(q^(111/20))]
[(2/5), 9*q^(1/5) + 94*q^(6/5) + 612*q^(11/5) + 2816*q^(16/5) + 10958*q^(21/5) + O(q^(26/5))]
[(1/2), q^(-1/4) - 2*q^(3/4) - 61*q^(7/4) - 518*q^(11/4) - 2677*q^(15/4) - 11280*q^(19/4) + O(q^(23/4))]
[(3/5), 9*q^(1/5) + 94*q^(6/5) + 612*q^(11/5) + 2816*q^(16/5) + 10958*q^(21/5) + O(q^(26/5))]
[(7/10), -31*q^(11/20) - 257*q^(31/20) - 1343*q^(51/20) - 5635*q^(71/20) - 20283*q^(91/20) + O(q^(111/20))]
[(4/5), q^(-1/5) + 6*q^(4/5) + 107*q^(9/5) + 698*q^(14/5) + 3436*q^(19/5) + 13644*q^(24/5) + O(q^(29/5))]
[(9/10), q^(-1/20) - 34*q^(19/20) - 253*q^(39/20) - 1381*q^(59/20) - 5879*q^(79/20) - 21615*q^(99/20) + O(q^(119/20))]
------------------------------------------------------------
[(0), 10 + 50*q + 260*q^2 + 1030*q^3 + 3500*q^4 + O(q^5)]
[(1/10), 6*q^(-1/20) + 16*q^(19/20) + 82*q^(39/20) + 304*q^(59/20) + 1046*q^(79/20) + 3120*q^(99/20) + O(q^(119/20))]
[(1/5), q^(-1/5) - 24*q^(4/5) - 143*q^(9/5) - 622*q^(14/5) - 2204*q^(19/5) - 6876*q^(24/5) + O(q^(29/5))]
[(3/10), -16*q^(11/20) - 102*q^(31/20) - 448*q^(51/20) - 1650*q^(71/20) - 5248*q^(91/20) + O(q^(111/20))]
[(2/5), -q^(1/5) + 14*q^(6/5) + 92*q^(11/5) + 386*q^(16/5) + 1318*q^(21/5) + O(q^(26/5))]
[(1/2), 20*q^(3/4) + 160*q^(7/4) + 700*q^(11/4) + 2560*q^(15/4) + 8000*q^(19/4) + O(q^(23/4))]
[(3/5), -q^(1/5) + 14*q^(6/5) + 92*q^(11/5) + 386*q^(16/5) + 1318*q^(21/5) + O(q^(26/5))]
[(7/10), -16*q^(11/20) - 102*q^(31/20) - 448*q^(51/20) - 1650*q^(71/20) - 5248*q^(91/20) + O(q^(111/20))]
[(4/5), q^(-1/5) - 24*q^(4/5) - 143*q^(9/5) - 622*q^(14/5) - 2204*q^(19/5) - 6876*q^(24/5) + O(q^(29/5))]
[(9/10), 6*q^(-1/20) + 16*q^(19/20) + 82*q^(39/20) + 304*q^(59/20) + 1046*q^(79/20) + 3120*q^(99/20) + O(q^(119/20))]
------------------------------------------------------------
[(0), 22 + 342*q + 2156*q^2 + 10258*q^3 + 39844*q^4 + O(q^5)]
[(1/10), -2*q^(-1/20) - 152*q^(19/20) - 1094*q^(39/20) - 5828*q^(59/20) - 24562*q^(79/20) - 89580*q^(99/20) + O(q^(119/20))]
[(1/5), 3*q^(-1/5) + 48*q^(4/5) + 571*q^(9/5) + 3414*q^(14/5) + 15948*q^(19/5) + 61452*q^(24/5) + O(q^(29/5))]
[(3/10), -108*q^(11/20) - 926*q^(31/20) - 4924*q^(51/20) - 20890*q^(71/20) - 75884*q^(91/20) + O(q^(111/20))]
[(2/5), 37*q^(1/5) + 362*q^(6/5) + 2356*q^(11/5) + 10878*q^(16/5) + 42514*q^(21/5) + O(q^(26/5))]
[(1/2), 4*q^(-1/4) - 28*q^(3/4) - 404*q^(7/4) - 2772*q^(11/4) - 13268*q^(15/4) - 53120*q^(19/4) + O(q^(23/4))]
[(3/5), 37*q^(1/5) + 362*q^(6/5) + 2356*q^(11/5) + 10878*q^(16/5) + 42514*q^(21/5) + O(q^(26/5))]
[(7/10), -108*q^(11/20) - 926*q^(31/20) - 4924*q^(51/20) - 20890*q^(71/20) - 75884*q^(91/20) + O(q^(111/20))]
[(4/5), 3*q^(-1/5) + 48*q^(4/5) + 571*q^(9/5) + 3414*q^(14/5) + 15948*q^(19/5) + 61452*q^(24/5) + O(q^(29/5))]
[(9/10), -2*q^(-1/20) - 152*q^(19/20) - 1094*q^(39/20) - 5828*q^(59/20) - 24562*q^(79/20) - 89580*q^(99/20) + O(q^(119/20))]
------------------------------------------------------------
[(0), 36 + 586*q + 3708*q^2 + 17694*q^3 + 68852*q^4 + O(q^5)]
[(1/10), -5*q^(-1/20) - 270*q^(19/20) - 1935*q^(39/20) - 10275*q^(59/20) - 43245*q^(79/20) - 157545*q^(99/20) + O(q^(119/20))]
[(1/5), 5*q^(-1/5) + 90*q^(4/5) + 1035*q^(9/5) + 6130*q^(14/5) + 28460*q^(19/5) + 109260*q^(24/5) + O(q^(29/5))]
[(3/10), -185*q^(11/20) - 1595*q^(31/20) - 8505*q^(51/20) - 36145*q^(71/20) - 131485*q^(91/20) + O(q^(111/20))]
[(2/5), 65*q^(1/5) + 630*q^(6/5) + 4100*q^(11/5) + 18940*q^(16/5) + 74070*q^(21/5) + O(q^(26/5))]
[(1/2), 7*q^(-1/4) - 54*q^(3/4) - 747*q^(7/4) - 5026*q^(11/4) - 23859*q^(15/4) - 94960*q^(19/4) + O(q^(23/4))]
[(3/5), 65*q^(1/5) + 630*q^(6/5) + 4100*q^(11/5) + 18940*q^(16/5) + 74070*q^(21/5) + O(q^(26/5))]
[(7/10), -185*q^(11/20) - 1595*q^(31/20) - 8505*q^(51/20) - 36145*q^(71/20) - 131485*q^(91/20) + O(q^(111/20))]
[(4/5), 5*q^(-1/5) + 90*q^(4/5) + 1035*q^(9/5) + 6130*q^(14/5) + 28460*q^(19/5) + 109260*q^(24/5) + O(q^(29/5))]
[(9/10), -5*q^(-1/20) - 270*q^(19/20) - 1935*q^(39/20) - 10275*q^(59/20) - 43245*q^(79/20) - 157545*q^(99/20) + O(q^(119/20))]
"""
S = self.gram_matrix()
w = self.weilrep()
wt = self.input_wt()
M, p, X = self._borcherds_product_polyhedron(pole_order,
prec,
verbose=verbose)
if verbose:
print('I will now try to find a Hilbert basis.')
try:
b = matrix(Cone(p).Hilbert_basis())
try:
u = M.solve_left(b)
Y = [v * X for v in u.rows()]
Y.sort(key=lambda x: x.fourier_expansion()[0][2][0])
Y = WeilRepModularFormsBasis(wt, Y, w)
except ValueError:
Y = WeilRepModularFormsBasis(wt, [], w)
except IndexError:
Y = WeilRepModularFormsBasis(wt, [], w)
if wt >= 0:
X = deepcopy(
w.basis_vanishing_to_order(wt, max(0, -pole_order), prec))
if X:
X.extend(Y)
return X
return Y
def cluster_fan(self, depth=infinity):
from sage.geometry.cone import Cone
from sage.geometry.fan import Fan
seeds = self.seeds(depth=depth, mutating_F=False)
cones = map(lambda s: Cone(s.g_vectors()), seeds)
return Fan(cones)