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def trinomiales(A, Vf, Vg, Vd, total=True):
r"""
Return data necessary to compute the zeta functions of
a triple singularity defined by :math:`z^3 -3 f z^2 + 2 g`, where
:math:`f, g` are polynomials in two variables.
We consider an embedded resolution of :math:`fg(f^3 - g^2)` with
a stratification of the exceptional divisor:
- :math:`\check{E}_i`: the smooth part of the total divisor
- :math:`E_i\cap E_j`: the double points of the divisor
- :math:`E_i\cap \{f=0\}`: the intersections with the strict
transform of :math:`f`
- :math:`E_i\cap \{g=0\}`: the intersections with the strict
transform of :math:`g`
- :math:`E_i\cap \{f^3=g^2\}`: the intersections with the strict
transform of :math:`f^3-g^2`
It is also possible to forget the strict transforms of :math:`f, g`
and even to consider only resolutions of :math:`f^3 - g^2`.
INPUT:
- ``A`` -- the incidence matrix of a resolution
- ``Vf`` -- the vector with the intersection numbers of the strict
transform of :math:`f` with the divisors of the resolution
- ``Vg`` -- the vector with the intersection numbers of the strict
transform of :math:`g` with the divisors of the resolution
- ``Vd`` -- the vector with the intersection numbers of the strict
transform of :math:`f^3 - g^2` with the divisors of the resolution
- ``total`` -- boolean (default: ``True``), whether we consider also
the strata corresponding to the strict transforms of :math:`f, g`.
OUTPUT:
Polynomials :math:`z^3 + M_1 z + M_0` or :math:`z^3 + M_2 z^2 + M_0` are
assigned to each strata. The monomials are of the form :math:`x^k` for the
smooth parts of the divisors and :math:`x^k y^l` for the double points.
Each variable is associated to a divisor or to a strict transform
- ``n`` -- the number of irreducible components of the exceptional divisor
- ``Nf`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
the total transform of :math:`f`
- ``Ng`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
the total transform of :math:`g`
- ``Nd`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
the total transform of :math:`f^3 - g^2`
- ``euler`` -- the vector in :math:`\mathbb{Z}^n` of the Euler
characteristic of the smooth part of the strata. If ``total=True`` this
smooth part does not contain the intersections with total transform
of :math:`f, g`
- ``nus`` -- the vector in :math:`\mathbb{Z}^n` of the canonical
multiplicities
- ``pols`` -- a dictionary with keys ``(i, )``, ``(i, j)``, ``(i, 'f')``
(if ``total=True``), ``(i, 'g')`` (if ``total=True``), and ``(i, 'd')``
assigning the polynomial of the stratum (non-degenerated with respect
to its Newton polytope), the weights ``(nui, nuj, 1)`` coding
the form
:math:`x^{\nu_i-1}y^{\nu_j-1}\operatorname{d}\! x \operatorname{d}\! y \operatorname{d}\! z`
EXAMPLES::
sage: R.<x, y, z> = QQ[]
sage: f, g = (x, y)
sage: A = Matrix(QQ, 3, [-3, 0, 1, 0, -2, 1, 1, 1, -1])
sage: Vf = vector(QQ, [1, 0, 0])
sage: Vg = vector(QQ, [0, 1, 0])
sage: Vd = vector(QQ, [0, 0, 1])
sage: trinomiales(A, Vf, Vg, Vd)
(3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (0, 0, -1), (2, 3, 5),
{(0, 'f'): (x + x*y*z + z^3, (2, 1, 1), 1),
(0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
(1, 'g'): (x*z + x^2*y + z^3, (3, 1, 1), 1),
(1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
(2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
0: (x + x*z + z^3, (2, 1, 1), 0),
1: (x^2 + x*z + z^3, (3, 1, 1), 0),
2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
sage: trinomiales(A, Vf, Vg, Vd, total=False)
(3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (1, 1, -1), (2, 3, 5),
{(0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
(1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
(2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
0: (x + x*z + z^3, (2, 1, 1), 1),
1: (x^2 + x*z + z^3, (3, 1, 1), 1),
2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
"""
R = PolynomialRing(QQ, 'x, y, z', order='negdeglex')
x, y, z = R.gens()
n = A.nrows()
A1 = -A ** -1
Nf, Ng, Nd = [A1 * N0 for N0 in (Vf, Vg, Vd)]
diag = vector(A.diagonal())
unos = vector(n * [1])
if total:
VV = Vf + Vg
else:
VV = Vf.parent()(0)
euler = (-A + 2 * identity_matrix(n)) * unos + diag - VV - Vd
nus = A1 * (diag + 2 * unos) + unos
expos = {}
for j in range(n):
a, b = (Nf[j], Ng[j])
eq = 3 * a == 2 * Ng[j]
nnd = min(3 * a, 2 * b) == Nd[j]
if nnd:
expos[j] = ((a, b), (a / 2, b), nus[j], eq, nnd)
else:
expos[j] = ((a, b), (a / 2, Nd[j] - b), nus[j], eq, nnd)
cadenas = ('d',)
if total:
expos['f'] = ((1, 0), (0, 0), 1, False, True)
expos['g'] = ((0, 1), (0, 0), 1, False, True)
cadenas = ('f', 'g') + cadenas
expos['d'] = ((0, 0), (0, 1), 1, True, False)
pols = {}
for j in range(n):
if expos[j][-1]:
a, b = expos[j][0]
pols[j] = (z ** 3 + x ** a * z + x ** b, (expos[j][2], 1, 1),
euler[j])
else:
a, b = expos[j][1]
pols[j] = (z ** 3 + x ** a * z ** 2 + x ** b,
(expos[j][2], 1, 1), euler[j])
interdiv = [(i, j) for i, j in Combinations(n, 2) if A[i, j] != 0]
interstrict = [(i, j) for i in range(n) for j in cadenas
if eval('V' + j)[i] != 0]
for i, j in interdiv + interstrict:
if j in ZZ:
ep = 1
else:
ep = eval('V' + j)[i]
eqi, nndi = expos[i][-2:]
eqj, nndj = expos[j][-2:]
nd = eqi and eqj and not (nndi and nndj)
if not nd:
ai, bi = expos[i][0]
aj, bj = expos[j][0]
pols[i, j] = (z ** 3 + x ** ai * y ** aj * z + x ** bi * y ** bj,
(expos[i][2], expos[j][2], 1), ep)
else:
ai, bi = expos[i][1]
aj, bj = expos[j][1]
pols[i, j] = (z ** 3 + x ** ai * y ** aj * z ** 2 +
x ** bi * y ** bj, (expos[i][2], expos[j][2], 1), ep)
return n, Nf, Ng, Nd, euler, nus, pols
def estrato(pol, pesos, euler, info=False):
r"""
Return the topological and Igusa zeta functions associated to a stratum
INPUT:
- ``pol`` -- a trinomial non degenerated with respect to its Newton
polyhedron
- ``pesos`` -- the canonical weights :math:`\nu:i`
- `euler`` -- the Euler characteristic of the stratum
- ``info`` -- boolean (default: False), if ``True`` a dictionary giving
the powers of cyclotomic polynomials for the monodromy zeta function
is given
EXAMPLES::
sage: R.<x, y, z> = QQ[]
sage: F = z^3 + x^4 * y^6 * z + x^7 * y^9
sage: estrato(F, (1, 1, 1), 1, info=True)
((1/3*s^2 + 1/2*s + 2/9)/(s^3 + 35/18*s^2 + 7/6*s + 2/9), -t^3 + 1,
{1: 1, 3: 1})
"""
Z1 = ZetaFunctions(pol)
zetatop = euler * Z1.topological_zeta(local=True, weights=pesos)
zetamon = Z1.monodromy_zeta() ** euler
if info:
dic = {}
for p, j in zetamon.factor():
c = p.is_cyclotomic(certificate=True)
dic[c] = j
return zetatop, zetamon, dic
return zetatop, zetamon
def zetas(n, pols, info=False):
r"""
Return the topological and Igusa zeta functions from the data of
a stratification
INPUT:
- ``n`` -- the number of exceptional divisors
- ``pols`` -- a dictionary from ``trinomiales``
- ``info`` -- boolean (default: False), if ``True`` a dictionary giving
the powers of cyclotomic polynomials for the monodromy zeta function
is given
EXAMPLES::
sage: A = Matrix(QQ, 3, [-3, 0, 1, 0, -2, 1, 1, 1, -1])
sage: Vf = vector(QQ, [1, 0, 0])
sage: Vg = vector(QQ, [0, 1, 0])
sage: Vd = vector(QQ, [0, 0, 1])
sage: n, Nf, Ng, Nd, euler, nus, pols = trinomiales(A, Vf, Vg, Vd)
sage: zetas(n, pols, info=True)
(1/(s + 1), -t + 1, {1: 1})
"""
zetatop = 0
T = PolynomialRing(QQ, 't')
t = T.gen(0)
zetamon = 1 / (1 - t ** 3) ** n
for k in pols:
zt, zm = estrato(*pols[k])
zetatop += zt
zetamon *= zm
if info:
dic = {}
for p, j in zetamon.factor():
c = p.is_cyclotomic(certificate=True)
dic[c] = j
return zetatop, zetamon, dic
return zetatop, zetamon

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zetas_triples.py