def root(self):
if self.mode == 'K':
ell = self.ell
self.RS = RationalSet([PositiveOrthant(ell)], ambient_dim=ell)
actual_ring = self.ring
F = FractionField(actual_ring)
r = matrix(F, self.R).rank()
self.r = r
if not self.d:
return
F = [
LaurentIdeal(
gens=[LaurentPolynomial(f) for f in self.R.minors(j)],
RS=self.RS,
normalise=True) for j in range(r + 1)
]
elif self.mode == 'O':
self.RS = RationalSet([PositiveOrthant(self.d)],
ambient_dim=self.d)
actual_ring = PolynomialRing(QQ, 'x', self.d)
xx = vector(actual_ring, actual_ring.gens())
self.C = matrix(actual_ring,
[xx * matrix(actual_ring, A) for A in self.basis])
r = matrix(FractionField(actual_ring), self.C).rank()
self.r = r
if not self.d:
return
F = [
LaurentIdeal(
gens=[LaurentPolynomial(f) for f in self.C.minors(j)],
RS=self.RS,
normalise=True) for j in range(r + 1)
]
else:
raise ValueError('invalid mode')
oo = r + 1
# On pairs:
# The first component is used as is, the second is multiplied by the extra
# variable. Note that index 0 corresponds to {1} and index oo to {0}.
self.pairs = ([(oo, 0)] + [(i, oo) for i in range(1, r)] +
[(i, i - 1) for i in range(1, r + 1)])
# Total number of pairs: 2 * r
self.integrand = ((1, ) + (2 * r - 1) * (0, ),
(self.d - r + 1, ) + (r - 1) * (-1, ) + r * (+1, ))
self.datum = IgusaDatum(F + [
LaurentIdeal(gens=[], RS=self.RS, ring=FractionField(actual_ring))
]).simplify()
return self.datum
def __init__(self, *polynomials):
if not polynomials:
raise ValueError('need at least one polynomial')
polynomials = [f if isinstance(f, LaurentPolynomial) else LaurentPolynomial(FractionField(f.parent())(f)) for f in polynomials]
self.nvars = polynomials[0].nvars
self.ideal = LaurentIdeal(gens=polynomials, RS=RationalSet(PositiveOrthant(self.nvars)), normalise=True)
def split_off_torus(F):
# Given a list F of polynomials, return G,d,T where len(F) == len(G)
# and G consists of polynomials in d variables; T = change of basis.
# There exists A in GL(n,ZZ) s.t. F[i]^A == G[i] * (Laurent term).
# In particular, F is non-degenerate relative to {0} iff this the case
# for G; cf Remark 4.3(ii) and Lemma 6.1(ii) in arXiv:1405.5711.
#
# The number of variables in G is the dimension of Newton(prod(F)).
#
if not F:
return [], 0, None
R = F[0].parent()
v = {f: vector(ZZ, f.exponents()[0]) for f in F}
submodule = (ZZ**R.ngens()).submodule(
chain.from_iterable(
(vector(ZZ, e) - v[f] for e in f.exponents()) for f in F))
_, _, T = matrix(ZZ, submodule.basis_matrix()).smith_form()
d = submodule.rank()
Rd = PolynomialRing(QQ, d, R.gens()[:d])
K = FractionField(R)
return [
Rd(
normalise_laurent_polynomial(
K(f([monomial_exp(K, e) for e in T.rows()])))) for f in F
], d, T
def _purge_multiples(self):
if self.is_empty():
return self
D = conify_polyhedron(self.polyhedron)
F = FractionField(self.ring)
def lte(i, j):
if i == j:
return True
rat = F(self.rhs[j] * self.lhs[i] / self.rhs[i] / self.lhs[j])
try:
alpha = monomial_log(rat.numerator()) - monomial_log(
rat.denominator())
except ValueError:
return False
return is_contained_in_dual(Cone([alpha]), D)
mins = minimal_elements(range(len(self.rhs)), lte)
if len(mins) == len(self.rhs):
return self
return ToricDatum(self.ring,
self.integrand,
[(self.lhs[i], self.rhs[i]) for i in mins],
[self.initials[i] for i in mins],
self.polyhedron,
depth=self._depth)
def normalise_laurent_polynomial(f):
"""
Rescale a Laurent polynomial by Laurent monomials. Details TBA.
"""
# Since Sage's Laurent polynomials are useless, we just use rational
# functions instead.
# First make sure, 'f' really is one.
R = f.parent()
K = FractionField(R)
f = K(f)
if K.ngens() == 0:
return R(0) if not f else R(1)
f = f.numerator()
return R(f / gcd(f.monomials()))
def common_overring(R, S):
if R.has_coerce_map_from(S):
return R
elif S.has_coerce_map_from(R):
return S
else:
combined_vars = list(
set(R.base_ring().gens()) | set(S.base_ring().gens()))
K = FractionField(PolynomialRing(QQ, len(combined_vars),
combined_vars))
return PolynomialRing(K, R.gens())
def make_map_latex(map_str):
# FIXME: Get rid of nu when map is defined over QQ
if "nu" not in map_str:
R0 = QQ
else:
R0 = PolynomialRing(QQ, 'nu')
R = PolynomialRing(R0, 2, 'x,y')
F = FractionField(R)
phi = F(map_str)
num = phi.numerator()
den = phi.denominator()
c_num = num.denominator()
c_den = den.denominator()
lc = c_den / c_num
# rescale coeffs to make them integral. then try to factor out gcds
# numerator
num_new = c_num * num
num_cs = num_new.coefficients()
if R0 == QQ:
num_cs_ZZ = num_cs
else:
num_cs_ZZ = []
for el in num_cs:
num_cs_ZZ = num_cs_ZZ + el.coefficients()
num_gcd = gcd(num_cs_ZZ)
# denominator
den_new = c_den * den
den_cs = den_new.coefficients()
if R0 == QQ:
den_cs_ZZ = den_cs
else:
den_cs_ZZ = []
for el in den_cs:
den_cs_ZZ = den_cs_ZZ + el.coefficients()
den_gcd = gcd(den_cs_ZZ)
lc = lc * (num_gcd / den_gcd)
num_new = num_new / num_gcd
den_new = den_new / den_gcd
# make strings for lc, num, and den
num_str = latex(num_new)
den_str = latex(den_new)
if lc == 1:
lc_str = ""
else:
lc_str = latex(lc)
if den_new == 1:
if lc == 1:
phi_str = num_str
else:
phi_str = lc_str + "(" + num_str + ")"
else:
phi_str = lc_str + "\\frac{" + num_str + "}" + "{" + den_str + "}"
return phi_str
def make_curve_latex(crv_str):
# FIXME: Get rid of nu when map is defined over QQ
if "nu" not in crv_str:
R0 = QQ
else:
R0 = PolynomialRing(QQ, 'nu')
R = PolynomialRing(R0, 2, 'x,y')
F = FractionField(R)
sides = crv_str.split("=")
lhs = latex(F(sides[0])).replace('(', '\\left(').replace(')', '\\right)')
rhs = latex(F(sides[1])).replace('(', '\\left(').replace(')', '\\right)')
eqn_str = lhs + '=' + rhs
return eqn_str
def make_curve_latex(crv_str):
# FIXME: Get rid of nu when map is defined over QQ
if "nu" not in crv_str:
R0 = QQ
else:
R0 = PolynomialRing(QQ, "nu")
R = PolynomialRing(R0, 2, "x,y")
F = FractionField(R)
sides = crv_str.split("=")
lhs = latex(F(sides[0]))
rhs = latex(F(sides[1]))
eqn_str = lhs + "=" + rhs
return eqn_str
def root(self):
d = self.ring.ngens()
self.d = d
polyhedra = []
for i in range(d):
eqns = [
(i+1) * [0] + [1] + (d-i-1) * [0]
]
ieqs = [
[-1] + j * [0] + [1] + (d-j-1) * [0]
for j in range(i)
]
ieqs += [
(j+1) * [0] + [1] + (d-j-1) * [0]
for j in range(i+1,d)
]
polyhedra.append(Polyhedron(eqns=eqns, ieqs=ieqs))
self.RS = RationalSet(polyhedra, ambient_dim=d)
# NOTE:
# A bug in Sage prevents rank computations over (fields of fractions of)
# polynomial rings in zero variables over fields.
if d > 0:
F = FractionField(self.ring)
else:
F = FractionField(self.ring.base_ring())
two_u = matrix(F, self.R).rank() # self.R.rank()
if two_u % 2:
raise RuntimeError('this is odd')
self.u = two_u // 2
self.v = matrix(F, self.S).rank() # self.S.rank()
if not d:
return
F = [
LaurentIdeal(
gens = [LaurentPolynomial(_sqrt(f)) for f in principal_minors(self.R, 2*j)],
RS = self.RS,
normalise = True)
for j in range(0, self.u+1)
]
G = [LaurentIdeal(gens=[LaurentPolynomial(g) for g in self.S.minors(j)],
RS=self.RS,
normalise=True)
for j in range(self.v + 1)]
oo = self.u + self.v + 2
# On pairs:
# The first component is used as is, the second is multiplied by the extra
# variable. Note that index 0 corresponds to {1} and index oo to {0}.
# We skip the |F_1|/|F_0 cap xF_1| factor which is generically trivial.
self.pairs = (
[(oo, 0)] +
[(i, oo) for i in range(2, self.u)] + [(i + 1, i) for i in range(1, self.u)] +
[(i, oo) for i in range(self.u + 2, self.u + self.v + 1)] +
[(i + 1, i) for i in range(self.u + 1, self.u + self.v + 1)])
# Note: q^b t^a really corresponds to a*s - b, in contrast to subobjects, where
# the (-1)-shift coming from Jacobians is included.
# This also means we don't have to manually add (-1)s for extra variables.
self.integrand = (
(self.u,) + (self.u - 2) * (1,) + (self.u - 1) * (-1,) + (2 * self.v - 1) * (0,),
(self.d + 1 - self.v,) + (2 * self.u - 3) * (0,) + (self.v - 1) * (-1,) + self.v * (1,))
self.datum = IgusaDatum(F + G + [LaurentIdeal(gens=[], RS=self.RS, ring=FractionField(self.ring))])
self.datum = self.datum.simplify()
return self.datum
def make_map_latex(map_str, nu=None):
if "nu" not in map_str:
R0 = QQ
else:
R0 = PolynomialRing(QQ, "nu")
R = PolynomialRing(R0, 2, "x,y")
F = FractionField(R)
phi = F(map_str)
num = phi.numerator()
den = phi.denominator()
c_num = num.denominator()
c_den = den.denominator()
lc = c_den / c_num
# rescale coeffs to make them integral. then try to factor out gcds
# numerator
num_new = c_num * num
num_cs = num_new.coefficients()
if R0 == QQ:
num_cs_ZZ = num_cs
else:
num_cs_ZZ = []
for el in num_cs:
num_cs_ZZ = num_cs_ZZ + el.coefficients()
num_gcd = gcd(num_cs_ZZ)
# denominator
den_new = c_den * den
den_cs = den_new.coefficients()
if R0 == QQ:
den_cs_ZZ = den_cs
else:
den_cs_ZZ = []
for el in den_cs:
den_cs_ZZ = den_cs_ZZ + el.coefficients()
den_gcd = gcd(den_cs_ZZ)
lc = lc * (num_gcd / den_gcd)
num_new = num_new / num_gcd
den_new = den_new / den_gcd
# evaluate at nu, if given
if nu and ("nu" in map_str):
S = PolynomialRing(CC, 2, 'x,y')
lc = lc.subs(nu=nu)
num_dict = dict()
den_dict = dict()
for m, c in num_new.dict().items():
num_dict[m] = c.subs(nu=nu)
for m, c in den_new.dict().items():
den_dict[m] = c.subs(nu=nu)
num_new = S(num_dict)
den_new = S(den_dict)
# make strings for lc, num, and den
num_str = latex(num_new)
den_str = latex(den_new)
if lc == 1:
lc_str = ""
else:
lc_str = latex(lc)
if den_new == 1:
if lc == 1:
phi_str = num_str
else:
phi_str = lc_str + "(" + num_str + ")"
else:
phi_str = lc_str + "\\frac{%s}{%s}" % (num_str, den_str)
return phi_str
def from_polyhedron(cls, P, ring, base_list=None):
"""
Use LattE to compute the generating function of a rational polyhedron
as a sum of small rational functions.
"""
if P.is_empty():
if ring.ngens() != P.ambient_dim():
raise TypeError('Dimension mismatch')
return cls(ring, base_list=base_list)
elif P.is_zero():
return cls([CyclotomicRationalFunction(ring.one())],
base_list=base_list)
elif len(P.vertices()) == 1 and (not P.rays()) and (not P.lines()):
# For some reason, LattE doesn't produce .rat files for points.
return cls([
CyclotomicRationalFunction(
ring.one(), exponents=[vector(ZZ,
P.vertices()[0])])
])
hrep = 'polyhedron.hrep'
ratfun = hrep + '.rat'
with TemporaryDirectory() as tmpdir, cd(tmpdir):
with open(hrep, 'w') as f:
f.write(latteify_polyhedron(P))
with open('/dev/null', 'w') as DEVNULL:
retcode = subprocess.call([
common.count, '--compute-vertex-cones=4ti2',
'--triangulation=cddlib',
'--multivariate-generating-function', hrep
],
stdout=DEVNULL,
stderr=DEVNULL,
env=augmented_env(common.count))
if retcode != 0:
raise RuntimeError(
'LattE failed. Make sure it has been patched in order to be compatible with Zeta.'
)
K = FractionField(ring)
variables = [K.coerce(x) for x in ring.gens()]
def exp(a):
return K.prod(x**e for x, e in zip(variables, a))
def vectorise_string(s):
return vector(ZZ, s.strip().split())
with TemporaryList() as summands, open(ratfun, 'r') as f:
while True:
line = f.readline()
if not line:
break
line = line.strip()
if not line:
continue # ignore empty lines
# The modified version of 'count' produces files in the following format:
# {
# scalar
# nterms
# a[1] ... a[n] \
# ... | nterms many
# c[1] ... c[n] /
# nrays
# u[1] ... u[n] \
# ... | nrays many
# w[1] ... w[n] /
# }
# ...
# This corresponds to scalar * sum(X^a + ... + X^c) / (1 - X^u) / ... / (1-X^w).
if line != "{":
raise RuntimeError(
'Invalid LattE output (BEGIN) [line=%s]' % line)
scalar = ring(f.readline())
nterms = int(f.readline())
numerator = K(scalar) * K.sum(
exp(vectorise_string(f.readline()))
for _ in range(nterms))
nrays = int(f.readline())
exponents = [vector(ZZ, len(variables))] + [
vectorise_string(f.readline()) for _ in range(nrays)
]
line = f.readline().strip()
if line != '}':
raise RuntimeError('Invalid Latte output (END)')
summands.append(
CyclotomicRationalFunction.from_laurent_polynomial(
numerator, ring, exponents))
return cls(summands, base_list=base_list)
