Steenbrink/Suspension10.ipynb

In [1]:

R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')
R2.<u, v> = PolynomialRing(QQ, order='negdeglex')
R1.<s> = QQ[]

In [3]:

f = u^5 + v^6
F = z^10 + f(u=x, v=y)
Z = ZetaFunctions(F)

In [37]:

zt = Z.topological_zeta(local=True)

In [14]:

g0 = v^10 - u^30
Z0 = ZetaFunctions(g0)
zt0 = Z0.topological_zeta(local=True, weights=[11, 1])

In [15]:

g1 = v^10 - u^6
Z1 = ZetaFunctions(g1)
zt1 = Z1.topological_zeta(local=True, weights=[11 / 5, 1])

In [16]:

g2 = v^10 - u^5
Z2 = ZetaFunctions(g2)
zt2 = Z2.topological_zeta(local=True, weights=[11 / 6, 1])

In [17]:

g3 = z^10 - x^30 * y
Z3 = ZetaFunctions(g3)
zt3 = Z3.topological_zeta(local=True, weights=[11, 1, 1])

Out[17]:

(1/110) * (s + 7/15)^-1 * (s + 1)^-1 * (s + 14/3)

In [19]:

zt - (-zt0 + zt1 + zt2 + zt3)

Out[19]:

0

In [20]:

zt10 = Z.topological_zeta(local=True, d=10)
zt10.factor()

Out[20]:

(-1/6) * (s + 7/15)^-1

In [22]:

zt5 = Z.topological_zeta(local=True, d=5) - zt10
zt5

Out[22]:

1/5/(s + 7/15)

In [23]:

zt2 = Z.topological_zeta(local=True, d=2) - zt10
zt2

Out[23]:

1/6/(s + 7/15)

In [25]:

Zf = ZetaFunctions(f)

In [28]:

ztf = Zf.topological_zeta(local=True)
ztf10 = Zf.topological_zeta(local=True, d=10)
ztf5 = Zf.topological_zeta(local=True, d=5) - ztf10
ztf2 = Zf.topological_zeta(local=True, d=2) - ztf10

In [34]:

p = s + 1 / 10

In [47]:

h = {}
h[1, 0] = 1 / 10 / p
h[1, 1] = 9 / 10 * s / (s + 1) * (p + 1) / p
h[1, 2] = 3 / 10 * s / (s + 1)
h[1, 5] = 24 / 10 * s / (s + 1)
h[1, 10] = 99 / 10 * s / (s + 1) 
aux =  h[1, 0] +  h[1, 1] * ztf(s=p) -  h[1, 2] * ztf2(s=p) -  h[1, 5] * ztf5(s=p) -  h[1, 10] * ztf10(s=p)

In [48]:

aux - zt

Out[48]:

0

In [63]:

h[2, 0] = 1 / 10 / p
h[2, 1] = 1 / 10 * (p + 1) / p
h[2, 2] = 7 / 10
h[2, 5] = -14 / 10
h[2, 10] = -89 / 10
aux =  h[2, 0] +  h[2, 1] * ztf(s=p) +  h[2, 2] * ztf2(s=p) +  h[2, 5] * ztf5(s=p) +  h[2, 10] * ztf10(s=p)

In [64]:

(aux - zt2).factor()

Out[64]:

(1/10) * (s + 1/10)^-1 * (s + 7/15)^-1 * (s + 5/6)

In [ ]: