Steenbrink/resolucionlarga.ipynb

In [1]:

S.<x, y, z> = PolynomialRing(QQ, order='neglex')
f=((y^2+x^3)^2 + x^5*y)*((x^2+y^3)^2 + y^5*x)
g=(y^6+x^7)*(x^6+y^7)
h = f^3 - g^2
F= z^3 - 3 * f * z + 2 * g

In [2]:

h = f^3 - g^2
F.discriminant(z) - 108 * h

Out[2]:

0

In [3]:

D=F.discriminant(z)
# load('ZetaFunctionsNewtonND.py')
ZD = ZetaFunctions(D)
ZD.is_newton_non_degenerated(local=True)

Out[3]:

True

In [4]:

for p in ZD.get_polyfaces_dictionary(compact=True):
    print(p.factor())

(-108) * y^26
(-216) * x^6 * y^19
(-108) * x^26
(-216) * y^6 * x^19
(-108) * y^19 * (y^7 + 2*x^6)
(-216) * (y + x) * x^6 * y^6 * (y^12 - x*y^11 + x^2*y^10 - x^3*y^9 - 2*x^4*y^8 + 2*x^5*y^7 - 2*x^6*y^6 + 2*x^7*y^5 - 2*x^8*y^4 - x^9*y^3 + x^10*y^2 - x^11*y + x^12)
(-108) * x^19 * (2*y^6 + x^7)

In [5]:

for p in ZetaFunctions(g).get_polyfaces_dictionary(compact=True):
    print(p.factor())

y^13
x^6 * y^6
x^13
y^6 * (y^7 + x^6)
x^6 * (y^6 + x^7)

In [6]:

for p in ZetaFunctions(f).get_polyfaces_dictionary(compact=True):
    print(p.factor())

y^10
x^10
x^4 * y^4
y^4 * (y^3 + x^2)^2
x^4 * (y^2 + x^3)^2

In [9]:

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

Veamos que la resolución larga no aporta

In [10]:

M39, Z39 = estrato(z^3+x^20*y^42*z+x^26*y^52, 1, [5,11,1])
M98, Z98 = estrato(z^3+x^42*y^21*z+x^52*y^26, 1, [11,6,1])
M38, Z38 = estrato(z^3+x^20*y^21*z+x^26*y^26, 1, [5,6,1])

In [12]:

Z39 + Z98 - Z38

Out[12]:

0

In [13]:

estrato(z^3+x^20*y^42*z+x^26*y^52, 1, [5,11,1]) == estrato(z^3+0*x^20*y^42*z+x^26*y^52, 1, [5,11,1])

Out[13]:

True

In [14]:

M3, Z3 = estrato(z^3+x^20*z+x^26, 1, [5,1,1])
M8, Z8 = estrato(z^3+x^21*z+x^26, 1, [6,1,1])

In [15]:

Z38 + Z8 - Z3

Out[15]:

0

In [39]:

a, b = estrato(z^3+x^4*z^2+x^13, 1, [2,1,1])
a.factor(), b.factor()

Out[39]:

((t + 1)^-2 * (t^2 - t + 1)^-1 * (t^2 + 1)^-1 * (t^4 - t^2 + 1)^-1 * (t^12 - t^11 + t^10 - t^9 + t^8 - t^7 + t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^-1,
 (19/156) * (s + 1/2)^-2 * (s + 1)^-1 * (s^2 + 48/19*s + 39/38))

In [19]:

T.<s> = QQ[]

In [28]:

A1 = 3 * s / (s + 1) / (78 * s + 39)
V1 = 2 / (78 * s + 39)
V2 = 1 / (78 * s + 39)
M7, Z7 = estrato(z^3 + x^78, 1, [13,1,1])
Z7 - (V1 + V2 - A1)

Out[28]:

0

In [30]:

M7a, Z7a = estrato(z^3 + x^13, 1, [13/6,1,1])
Z7a.factor()

Out[30]:

(8/39) * (s + 1/2)^-1 * (s + 1)^-1 * (s + 9/8)

In [37]:

(6 * V1 + 6 * V2 - 2 / 3 * A1 - Z7a)

Out[37]:

0

Cálculo completo

In [9]:

M1, Zt1 = estrato(z^3+x^4*z^2 + x^13, -13, [2,1,1])
M1d, Zt1d = estrato(z^3+x^4*z^2 + x^13*y, 13, [2,1,1])
M17, Z17 = estrato(z^3+x^4*y^28*z^2 + x^13*y^78 , 2, [2,13,1])
M7, Z7 = estrato(z^3+x^4*z + x^13*y, -4, [13,1,1])
M7d, Z7d = estrato(z^3+x^4*z + x^13*y, -4, [13,1,1])
M7g, Z7g

  Cell In [9], line 3
    M17, Z17 = estrato(z**Integer(3)+, Integer(2), [Integer(2),Integer(13),Integer(1)])
                                     ^
SyntaxError: invalid syntax

In [55]:

G=z^3+x^4*z + x^13
ZG=ZetaFunctions(G)
ZG.give_info_newton(faces=False, cones=False, compact = False)
ZG.topological_zeta(local= True, weights = [2,1,1])

Newton's polyhedron of z^3 + x^4*z + x^13:
    support points = [(0, 0, 3), (4, 0, 1), (13, 0, 0)]
    vertices = [(0, 0, 3), (4, 0, 1), (13, 0, 0)]
    number of proper faces = 15
    Facet 0: z >= 0
    Facet 1: -13 + 9*z + x >= 0
    Facet 2: -6 + 2*z + x >= 0
    Facet 3: x >= 0
    Facet 4: y >= 0

Out[55]:

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

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