Steenbrink/Estratos.ipynb

In [18]:

T.<t> = QQ[]
S.<x, y, z> = PolynomialRing(QQ, order='neglex')
R.<u, v> = PolynomialRing(QQ, order='neglex')
f = u^2 * v^2 + u^5 + v^5
g = u^3 * v^3 + u^7 + v^7
F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)

In [19]:

F.jacobian_ideal().minimal_associated_primes()

Out[19]:

[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]

In [23]:

h = f^3 - g^2
F.discriminant(z)(x=u, y=v) - 108 * h

Out[23]:

0

In [22]:

F.jacobian_ideal().vector_space_dimension()

Out[22]:

65

In [24]:

h.jacobian_ideal().vector_space_dimension()

Out[24]:

148

In [27]:

load('/home/artal/Descargas/ZetaFunctionsNewtonND.py')

In [28]:

Z = ZetaFunctions(h)

In [47]:

[p.factor() for p in Z.get_polyfaces_dictionary(compact=True)]

Out[47]:

[(-1) * v^14,
 (-1) * u^14,
 (-2) * u^3 * v^10,
 (-2) * v^3 * u^10,
 (-1) * v^10 * (v^4 + 2*u^3),
 (-1) * u^10 * (2*v^3 + u^4),
 (-1) * (v + u) * u^3 * v^3 * (2*v^6 - 5*u*v^5 + 5*u^2*v^4 - 5*u^3*v^3 + 5*u^4*v^2 - 5*u^5*v + 2*u^6)]

In [46]:

Z.is_newton_degenerated(local=True)

Out[46]:

False

In [38]:

Z.newton_plot()

Out[38]:

In [49]:

Z.give_info_facets(compact=True)

    Facet 0: -42 + 4*v + 3*u >= 0
    Facet 1: -13 + v + u >= 0
    Facet 3: -42 + 3*v + 4*u >= 0

In [43]:

h1 = sum(h.monomial_coefficient(u^i * v^j) * u^i * v^j for i, j in h.exponents() if (i, j) == (14, 0) or (i, j) == (0, 14) or i + j == 13)
h1

Out[43]:

-v^14 - 2*u^3*v^10 + 3*u^4*v^9 + 3*u^9*v^4 - 2*u^10*v^3 - u^14

In [44]:

h1.jacobian_ideal().vector_space_dimension()

Out[44]:

148

In [50]:

h

Out[50]:

-v^14 + v^15 + 3*u^2*v^12 - 2*u^3*v^10 + 3*u^4*v^9 + 3*u^5*v^10 + 4*u^7*v^7 + 3*u^9*v^4 - 2*u^10*v^3 + 3*u^10*v^5 + 3*u^12*v^2 - u^14 + u^15

In [68]:

def estrato(ff, chi, pesos):
    Z1 = ZetaFunctions(ff)
    zetamon1 = Z1.monodromy_zeta().value()^chi
    zetatop1 = chi * Z1.topological_zeta(local=True, weights=pesos).value()
    return zetamon1, zetatop1

In [147]:

zetatop=0
zetamon=1

$\check{E}_1$

In [148]:

zetamon1, zetatop1 = estrato(z^3 + x^2 * z^2 + x^7, -7, [2, 1, 1])
zetamon *= zetamon1
zetatop += zetatop1
zetamon1.factor(), zetatop1.denominator().factor()

Out[148]:

((t + 1)^14 * (t^2 - t + 1)^7 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7,
 (s + 9/14) * (s + 1))

${E}_1\cap E_{4,i}$

In [149]:

zetamon14, zetatop14 = estrato(z^3 + x^2 * y^7 * z^2 + x^7 * y^21, 2, [2, 7, 1])
zetamon *= zetamon14
zetatop += zetatop14
zetamon14.factor(), zetatop14.denominator().factor()

Out[149]:

((t - 1)^2 * (t^2 + t + 1)^2, (s + 9/14) * (s + 2/3) * (s + 1))

${E}_1\cap\Delta$

In [150]:

zetamon1d, zetatop1d = estrato(z^3 + x^2 * z^2 + x^7 * y, 7, [2, 1, 1])
zetamon *= zetamon1d
zetatop += zetatop1d
zetamon1d.factor(), zetatop1d.denominator().factor()

Out[150]:

((t + 1)^-7 * (t^2 - t + 1)^-7, (s + 9/14) * (s + 1))

In [156]:

zetamone1= zetamon1*zetamon14*zetamon1d
zetatope1= zetatop1 + zetatop14 + zetatop1d

In [157]:

zetamone1.factor(), zetatope1.denominator().factor()

Out[157]:

((t - 1)^2 * (t + 1)^7 * (t^2 + t + 1)^2 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7,
 (s + 9/14) * (s + 2/3) * (s + 1))

$\check{E}_{2,i}$

In [153]:

zetamon2, zetatop2 = estrato(z^3 + x^5 * z + x^7, 2, [3, 1, 1])
zetamon *= zetamon2
zetatop += zetatop2
zetamon2.factor(), zetatop2.denominator().factor()

Out[153]:

((t - 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,
 (s + 16/21) * (s + 1))

In [154]:

cyclotomic_polynomial(21)

Out[154]:

x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1

${E}_{2,i}\cap E_{3,i}$

In [155]:

zetamon23, zetatop23= estrato(z^3 + x^5 * y^10 * z + x^7 * y^14, 2, [3, 5, 1])
zetamon *= zetamon23
zetatop += zetatop23
zetamon23.factor(), zetatop23.denominator().factor()

Out[155]:

((t - 1)^2 * (t^2 + t + 1)^2, (s + 29/42) * (s + 16/21) * (s + 1))

In [158]:

zetamone2= zetamon2*zetamon23
zetatope2= zetatop2 + zetatop23

In [159]:

zetamone2.factor(), zetatope2.denominator().factor()

Out[159]:

((t - 1)^4 * (t^2 + t + 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,
 (s + 29/42) * (s + 1))

$\check{E}_{3,i}$

In [160]:

zetamon3, zetatop3 = estrato(z^3 + x^10 * z + x^14, -2, [5, 1, 1])
zetamon *= zetamon3
zetatop += zetatop3
zetamon3.factor(), zetatop3.denominator().factor()

Out[160]:

((t - 1)^-2 * (t^2 - t + 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^2 * (t^12 + t^11 - t^9 - t^8 + t^6 - t^4 - t^3 + t + 1)^2,
 (s + 29/42) * (s + 1))

In [161]:

cyclotomic_polynomial(42)

Out[161]:

x^12 + x^11 - x^9 - x^8 + x^6 - x^4 - x^3 + x + 1

${E}_{3,i}\cap E_{4,i}$

In [162]:

zetamon34, zetatop34 = estrato(z^3 + x^14 * y^10 * z + x^21 * y^14, 2, [7, 5, 1])
zetamon *= zetamon34
zetatop += zetatop34
zetamon34.factor(), zetatop34.denominator().factor()

Out[162]:

((t - 1)^2 * (t^2 + t + 1)^2, (s + 2/3) * (s + 29/42) * (s + 1))

${E}_{3,i}\cap f$

In [163]:

zetamon3f, zetatop3f = estrato(z^3 + x * y^10 * z + y^14, 2, [1, 5, 1])
zetamon *= zetamon3f
zetatop += zetatop3f
zetamon3f.factor(), zetatop3f.denominator().factor()

Out[163]:

((t - 1)^2 * (t^2 - t + 1)^-2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2 * (t^12 + t^11 - t^9 - t^8 + t^6 - t^4 - t^3 + t + 1)^-2,
 (s + 29/42) * (s + 1))

In [164]:

zetamone23= zetamone2 * zetamon3 * zetamon34 * zetamon3f
zetatope23= zetatope2 + zetatop3 + zetatop34 + zetatop3f

In [165]:

zetamone23.factor(), zetatope23.denominator().factor()

Out[165]:

((t - 1)^6 * (t^2 + t + 1)^4 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,
 (s + 2/3) * (s + 1))

$\check{E}_{4,i}$

In [166]:

zetamon4, zetatop4 = estrato(z^3 + x^14 * z + x^21, -4, [7, 1, 1])
zetamon *= zetamon4
zetatop += zetatop4
zetamon4.factor(), zetatop4.denominator().factor()

Out[166]:

((t - 1)^4 * (t^2 + t + 1)^4 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^8 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^8,
 (s + 2/3) * (s + 1))

${E}_{4,i}\cap g$

In [167]:

zetamon4g, zetatop4g = estrato(z^3 + x^14 * z + x^21 * y, 2, [7, 1, 1])
zetamon *= zetamon4g
zetatop += zetatop4g
zetamon4g.factor(), zetatop4g.denominator().factor()

Out[167]:

((t - 1)^-2 * (t^2 + t + 1)^-2 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^-4 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-4,
 (s + 2/3) * (s + 1))

${E}_4\cap\Delta$

In [168]:

zetamon4d, zetatop4d = estrato(z^3 + x^7 * z^2 + x^21 * y, 2, [7, 1, 1])
zetamon *= zetamon4d
zetatop += zetatop4d
zetamon4d.factor(), zetatop4d.denominator().factor()

Out[168]:

((t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^-2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,
 (s + 2/3) * (s + 1))

In [172]:

zetamone4= zetamone23 * zetamon4 * zetamon4g * zetamon4d * zetamon14
zetatope4= zetatope23 + zetatop4 + zetatop4g + zetatop4d + zetatop14

In [173]:

zetamone4.factor(), zetatope4.denominator().factor()

Out[173]:

((t - 1)^10 * (t^2 + t + 1)^8 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^2,
 (s + 9/14) * (s + 2/3) * (s + 1))

In [143]:

zetamon /= (t^3 - 1)^7

In [144]:

zetamon

Out[144]:

t^66 - t^63 + 5*t^59 - 5*t^56 + 8*t^52 - 8*t^49 - 14*t^38 + 14*t^35 - 14*t^31 + 14*t^28 + 8*t^17 - 8*t^14 + 5*t^10 - 5*t^7 + t^3 - 1

In [145]:

zetatop.denominator().factor()

Out[145]:

(s + 9/14) * (s + 2/3) * (s + 1)

In [146]:

zetamon.factor()

Out[146]:

(t - 1)^3 * (t + 1)^7 * (t^2 + t + 1) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^2 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7

In [175]:

zetamon4 * zetamon4g^2

Out[175]:

1

In [176]:

zetamon3 * zetamon3f

Out[176]:

1

In [180]:

zetatope23.factor()

Out[180]:

(10/21) * (s + 2/3)^-1 * (s + 1)^-1 * (s + 6/5)

In [181]:

zetatope4.factor()

Out[181]:

(13/21) * (s + 9/14)^-1 * (s + 2/3)^-1 * (s + 1)^-1 * (s^2 + 43/26*s + 9/13)

In [ ]: