Steenbrink/resolucionlarga.ipynb

In [3]:

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 [4]:

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

Out[4]:

0

In [5]:

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

Out[5]:

False

In [6]:

ZD.give_info_newton(faces=False, cones=False, compact = False)

Newton's polyhedron of -108*y^26 + 108*y^30 + 324*x*y^29 + 648*x^2*y^27 + 324*x^2*y^28 + 1296*x^3*y^26 + 108*x^3*y^27 + 648*x^3*y^28 + 1620*x^4*y^24 + 648*x^4*y^25 + 1944*x^4*y^27 + 1944*x^5*y^23 + 3888*x^5*y^25 + 1944*x^5*y^26 + 324*x^5*y^27 - 216*x^6*y^19 + 2160*x^6*y^21 + 324*x^6*y^22 + 7776*x^6*y^24 + 648*x^6*y^25 + 2592*x^6*y^26 + 1080*x^7*y^20 + 9720*x^7*y^22 + 3888*x^7*y^23 + 1944*x^7*y^24 + 5832*x^7*y^25 + 1620*x^8*y^18 + 11664*x^8*y^21 + 13608*x^8*y^23 + 5184*x^8*y^24 + 1296*x^8*y^25 + 324*x^9*y^17 + 12960*x^9*y^19 + 1944*x^9*y^20 + 4860*x^9*y^21 + 21384*x^9*y^22 + 1620*x^9*y^23 + 6048*x^9*y^24 + 648*x^10*y^15 + 7776*x^10*y^18 + 30132*x^10*y^20 + 9720*x^10*y^21 + 7776*x^10*y^22 + 10368*x^10*y^23 + 324*x^10*y^24 + 9720*x^11*y^16 + 6480*x^11*y^18 + 30132*x^11*y^19 + 28512*x^11*y^21 + 7776*x^11*y^22 + 2916*x^11*y^23 + 1944*x^12*y^15 + 36288*x^12*y^17 + 4860*x^12*y^18 + 19440*x^12*y^19 + 33696*x^12*y^20 + 4104*x^12*y^21 + 8424*x^12*y^22 + 3456*x^13*y^13 + 4860*x^13*y^15 + 19440*x^13*y^16 + 55728*x^13*y^18 + 12960*x^13*y^19 + 15552*x^13*y^20 + 11016*x^13*y^21 + 648*x^13*y^22 + 25164*x^14*y^14 + 25920*x^14*y^16 + 42768*x^14*y^17 + 4860*x^14*y^18 + 34992*x^14*y^19 + 6804*x^14*y^20 + 3240*x^14*y^21 + 648*x^15*y^10 + 1944*x^15*y^12 + 4860*x^15*y^13 + 58752*x^15*y^15 + 6480*x^15*y^16 + 34992*x^15*y^17 + 31104*x^15*y^18 + 5508*x^15*y^19 + 6480*x^15*y^20 + 108*x^15*y^21 + 9720*x^16*y^11 + 19440*x^16*y^13 + 25920*x^16*y^14 + 6480*x^16*y^15 + 60264*x^16*y^16 + 9720*x^16*y^17 + 15552*x^16*y^18 + 6480*x^16*y^19 + 648*x^16*y^20 + 324*x^17*y^9 + 36288*x^17*y^12 + 42768*x^17*y^14 + 34992*x^17*y^15 + 9720*x^17*y^16 + 23328*x^17*y^17 + 3888*x^17*y^18 + 1620*x^17*y^19 + 1620*x^18*y^8 + 7776*x^18*y^10 + 6480*x^18*y^11 + 4860*x^18*y^12 + 55728*x^18*y^13 + 4860*x^18*y^14 + 31104*x^18*y^15 + 15552*x^18*y^16 + 3888*x^18*y^17 + 2160*x^18*y^18 - 216*x^19*y^6 + 12960*x^19*y^9 + 30132*x^19*y^11 + 19440*x^19*y^12 + 12960*x^19*y^13 + 34992*x^19*y^14 + 5508*x^19*y^15 + 6480*x^19*y^16 + 1620*x^19*y^17 + 1080*x^20*y^7 + 1944*x^20*y^9 + 30132*x^20*y^10 + 33696*x^20*y^12 + 15552*x^20*y^13 + 6804*x^20*y^14 + 6480*x^20*y^15 + 648*x^20*y^16 + 2160*x^21*y^6 + 11664*x^21*y^8 + 4860*x^21*y^9 + 9720*x^21*y^10 + 28512*x^21*y^11 + 4104*x^21*y^12 + 11016*x^21*y^13 + 3240*x^21*y^14 + 108*x^21*y^15 + 324*x^22*y^6 + 9720*x^22*y^7 + 21384*x^22*y^9 + 7776*x^22*y^10 + 7776*x^22*y^11 + 8424*x^22*y^12 + 648*x^22*y^13 + 1944*x^23*y^5 + 3888*x^23*y^7 + 13608*x^23*y^8 + 1620*x^23*y^9 + 10368*x^23*y^10 + 2916*x^23*y^11 + 1620*x^24*y^4 + 7776*x^24*y^6 + 1944*x^24*y^7 + 5184*x^24*y^8 + 6048*x^24*y^9 + 324*x^24*y^10 + 648*x^25*y^4 + 3888*x^25*y^5 + 648*x^25*y^6 + 5832*x^25*y^7 + 1296*x^25*y^8 - 108*x^26 + 1296*x^26*y^3 + 1944*x^26*y^5 + 2592*x^26*y^6 + 648*x^27*y^2 + 108*x^27*y^3 + 1944*x^27*y^4 + 324*x^27*y^5 + 324*x^28*y^2 + 648*x^28*y^3 + 324*x^29*y + 108*x^30:
    support points = [(0, 26, 0), (0, 30, 0), (1, 29, 0), (2, 27, 0), (2, 28, 0), (3, 26, 0), (3, 27, 0), (3, 28, 0), (4, 24, 0), (4, 25, 0), (4, 27, 0), (5, 23, 0), (5, 25, 0), (5, 26, 0), (5, 27, 0), (6, 19, 0), (6, 21, 0), (6, 22, 0), (6, 24, 0), (6, 25, 0), (6, 26, 0), (7, 20, 0), (7, 22, 0), (7, 23, 0), (7, 24, 0), (7, 25, 0), (8, 18, 0), (8, 21, 0), (8, 23, 0), (8, 24, 0), (8, 25, 0), (9, 17, 0), (9, 19, 0), (9, 20, 0), (9, 21, 0), (9, 22, 0), (9, 23, 0), (9, 24, 0), (10, 15, 0), (10, 18, 0), (10, 20, 0), (10, 21, 0), (10, 22, 0), (10, 23, 0), (10, 24, 0), (11, 16, 0), (11, 18, 0), (11, 19, 0), (11, 21, 0), (11, 22, 0), (11, 23, 0), (12, 15, 0), (12, 17, 0), (12, 18, 0), (12, 19, 0), (12, 20, 0), (12, 21, 0), (12, 22, 0), (13, 13, 0), (13, 15, 0), (13, 16, 0), (13, 18, 0), (13, 19, 0), (13, 20, 0), (13, 21, 0), (13, 22, 0), (14, 14, 0), (14, 16, 0), (14, 17, 0), (14, 18, 0), (14, 19, 0), (14, 20, 0), (14, 21, 0), (15, 10, 0), (15, 12, 0), (15, 13, 0), (15, 15, 0), (15, 16, 0), (15, 17, 0), (15, 18, 0), (15, 19, 0), (15, 20, 0), (15, 21, 0), (16, 11, 0), (16, 13, 0), (16, 14, 0), (16, 15, 0), (16, 16, 0), (16, 17, 0), (16, 18, 0), (16, 19, 0), (16, 20, 0), (17, 9, 0), (17, 12, 0), (17, 14, 0), (17, 15, 0), (17, 16, 0), (17, 17, 0), (17, 18, 0), (17, 19, 0), (18, 8, 0), (18, 10, 0), (18, 11, 0), (18, 12, 0), (18, 13, 0), (18, 14, 0), (18, 15, 0), (18, 16, 0), (18, 17, 0), (18, 18, 0), (19, 6, 0), (19, 9, 0), (19, 11, 0), (19, 12, 0), (19, 13, 0), (19, 14, 0), (19, 15, 0), (19, 16, 0), (19, 17, 0), (20, 7, 0), (20, 9, 0), (20, 10, 0), (20, 12, 0), (20, 13, 0), (20, 14, 0), (20, 15, 0), (20, 16, 0), (21, 6, 0), (21, 8, 0), (21, 9, 0), (21, 10, 0), (21, 11, 0), (21, 12, 0), (21, 13, 0), (21, 14, 0), (21, 15, 0), (22, 6, 0), (22, 7, 0), (22, 9, 0), (22, 10, 0), (22, 11, 0), (22, 12, 0), (22, 13, 0), (23, 5, 0), (23, 7, 0), (23, 8, 0), (23, 9, 0), (23, 10, 0), (23, 11, 0), (24, 4, 0), (24, 6, 0), (24, 7, 0), (24, 8, 0), (24, 9, 0), (24, 10, 0), (25, 4, 0), (25, 5, 0), (25, 6, 0), (25, 7, 0), (25, 8, 0), (26, 0, 0), (26, 3, 0), (26, 5, 0), (26, 6, 0), (27, 2, 0), (27, 3, 0), (27, 4, 0), (27, 5, 0), (28, 2, 0), (28, 3, 0), (29, 1, 0), (30, 0, 0)]
    vertices = [(0, 26, 0), (26, 0, 0), (19, 6, 0), (6, 19, 0)]
    number of proper faces = 19
    Facet 0: -156 + 7*y + 6*x >= 0
    Facet 1: -25 + y + x >= 0
    Facet 2: -156 + 6*y + 7*x >= 0
    Facet 3: x >= 0
    Facet 4: y >= 0
    Facet 5: z >= 0

In [7]:

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

Veamos que la resolución larga no aporta

In [8]:

M3, Z3 = estrato(z^3+x^26,-1, [5,1,1])
M9, Z9 = estrato(z^3+x^52,-1, [11,1,1])
M8, Z8 = estrato(z^3+x^26,1, [6,1,1])
M39, Z39 = estrato(z^3+x^26*y^52,1, [5,11,1])
M98, Z98 = estrato(z^3+x^52*y^26, 1, [11,6,1])
M9f, Z9f = estrato(z^3+x^50*y*z + x^52, 1, [11,1,1])
Z3 + Z39 + Z98 + Z8, Z9 + Z9f

Out[8]:

(0, 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)

In [0]:

In [0]: