Steenbrink/EjemplosGenerales.ipynb

In [79]:

load('zetas_triples_general.py')

In [19]:

R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')
R2.<u, v> = PolynomialRing(QQ, order='negdeglex')
def surface(f, g):
    sb = {u: x, v: y}
    return z^3 - 3 * f.subs(sb) * z + 2 * g.subs(sb)

def discrim3(f, g):
    return f^3 - g^2

def milnor(F):
    return F.jacobian_ideal().vector_space_dimension()

def factoresNewton(f):
    return [p.factor() for p in ZetaFunctions(f).get_polyfaces_dictionary(compact=True)
            if p.number_of_terms() > 1]

In [3]:

f = v^2 - u^3
g = v^3 - u^4
discrim3(f, g).factor(), milnor(surface(f, g))

Out[3]:

((-1) * u^3 * (-2*u*v^3 + 3*v^4 + u^5 - 3*u^3*v^2 + u^6), +Infinity)

In [150]:

H = ff^3 / gg^2
H1 = H(u=u * v^2)
H1(v=0, u=1/4).factor()

Out[150]:

-1 * 2^-8 * 3^3

In [127]:

f = u * v - u^3 - v^3
g = u^2 * v^2 - u^5 - v^5
hs = [(u, 1)]
ks = [(v, 2)]
ff = f * prod(p^j for p, j in hs) * prod(p for p, j in ks)
gg = g * prod(p^j for p, j in ks) * prod(p for p, j in hs)
d = R2(discrim3(ff, gg) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))
d.factor(), # milnor(surface(ff, gg))

Out[127]:

((-1) * (-u^4*v^3 + 3*u^6*v^2 + 3*u^3*v^5 - 3*u^8*v - 6*u^5*v^4 + u^4*v^5 - 3*u^2*v^7 + u^10 + u^7*v^3 + 3*u^4*v^6 - 2*u^2*v^8 + u*v^9 + u^10*v + 2*u^5*v^6 + v^11),)

In [128]:

factoresNewton(d)

Out[128]:

[(-1) * v^3 * (-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8), u^4 * (v - u^2)^3]

In [132]:

(-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8)(v=1).derivative(u).factor()

Out[132]:

(-1) * (-1 + 4*u) * (-1 + u)^2

In [143]:

(-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8)(v=1).derivative(u)(u=1 / 4)

Out[143]:

0

In [136]:

(ff(u=1 / 4, v=1)^3/gg(u=1 / 4, v=1)^2).factor()

Out[136]:

-1 * 7^6 * 31^-4

In [151]:

dd = R2(discrim3(-ff / 3, gg / 16) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))
factoresNewton(dd)

Out[151]:

[(-1/6912) * v^3 * (-4*u + v^2)^2 * (16*u^2 - 40*u*v^2 + 27*v^4),
 (-1/27) * u^4 * (v - u^2)^3]

In [153]:

factoresNewton(dd(u=u + v^2 / 4))

Out[153]:

[(1/6144) * v^7 * (-256*u^2 + 3*v^5),
 (-1/27) * u^4 * (v - u^2)^3,
 (-1/216) * u^2 * v^3 * (8*u^2 - 16*u*v^2 + 9*v^4)]

In [121]:

d1 = R2(d(v=v^2, u=v^3 * (u + 1)) / v^22)
g1 = R2(g(v=v^2, u=v^3 * (u + 1)) / v^10)

In [122]:

factoresNewton(d1)

Out[122]:

[-4*u^2 + v^5]

In [123]:

factoresNewton(g1)

Out[123]:

[(-1) * (-2*u + v^5)]

In [106]:

A = Matrix(QQ, 5)
for i, j in ((0, 2), (0, 4), (1, 2), (3, 4)):
    A[i, j] = A[j, i] = 1
A[0, 0] = -5
A[1, 1] = A[3, 3] = -2
A[2, 2] = A[4, 4] = -1
show(A)
Vf = vector(QQ, [0, 1, 0, 1, 0])
Vg = vector(QQ, [0, 0, 1, 0, 1])
Vd = vector(QQ, [5, 0, 1, 0, 1])
Vhs = ((vector(QQ, [0, 1, 0, 1, 0]), 1),)
Vks = ()

<html>\(\displaystyle \left(\begin{array}{rrrrr} -5 & 0 & 1 & 0 & 1 \\ 0 & -2 & 1 & 0 & 0 \\ 1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & -2 & 1 \\ 1 & 0 & 0 & 1 & -1 \end{array}\right)\)</html>

In [107]:

n, Nf, Ng, Nd, Nhs, Nks, euler, nus, pols == trinomiales_general(A, Vf, Vg, Vhs, Vks, Vd, total=True)

Out[107]:

(5,
 (2, 3, 5, 3, 5),
 (4, 5, 10, 5, 10),
 (9, 10, 20, 10, 20),
 [((1, 2, 3, 1, 2), 1), ((1, 1, 2, 2, 3), 1)],
 [],
 (-5, -1, -2, -1, -2),
 (2, 3, 5, 3, 5),
 False)

In [81]:

Nf + sum(j * V0 for V0, j in Nhs) + sum(V0 for V0, j in Nks)

Out[81]:

(4, 6, 10, 6, 10)

In [82]:

Ng + sum(j * V0 for V0, j in Nks) + sum(V0 for V0, j in Nhs)

Out[82]:

(6, 8, 15, 8, 15)

In [83]:

Nd + 3 * sum(V0 for V0, j in Nks) + 2 * sum(V0 for V0, j in Nhs)

Out[83]:

(13, 16, 30, 16, 30)

In [84]:

for k in pols:
    print(k)
    print(pols[k])
    print("===")

0
(z^3 + x^2*z^2 + x^7, (2, 1, 1), -5)
===
1
(z^3 + x^6*z + x^8, (3, 1, 1), -1)
===
2
(z^3 + x^10*z + x^15, (5, 1, 1), -2)
===
3
(z^3 + x^6*z + x^8, (3, 1, 1), -1)
===
4
(z^3 + x^10*z + x^15, (5, 1, 1), -2)
===
(0, 2)
(z^3 + x^2*y^5*z^2 + x^7*y^15, (2, 5, 1), 1)
===
(0, 4)
(z^3 + x^2*y^5*z^2 + x^7*y^15, (2, 5, 1), 1)
===
(1, 2)
(z^3 + x^6*y^10*z + x^8*y^15, (3, 5, 1), 1)
===
(3, 4)
(z^3 + x^6*y^10*z + x^8*y^15, (3, 5, 1), 1)
===
(0, 'd')
(z^3 + x^2*z^2 + x^7*y, (2, 1, 1), 5)
===
(1, 'f')
(z^3 + x^8 + x^6*y*z, (3, 1, 1), 1)
===
(2, 'g')
(z^3 + x^10*z + x^15*y, (5, 1, 1), 1)
===
(2, 'd')
(z^3 + x^5*z^2 + x^15*y, (5, 1, 1), 1)
===
(3, 'f')
(z^3 + x^8 + x^6*y*z, (3, 1, 1), 1)
===
(4, 'g')
(z^3 + x^10*z + x^15*y, (5, 1, 1), 1)
===
(4, 'd')
(z^3 + x^5*z^2 + x^15*y, (5, 1, 1), 1)
===
(1, ('h', 0))
(z^3 + x^6*y*z + x^8*y, (3, 1, 1), 1)
===
(3, ('h', 1))
(z^3 + x^6*y*z + x^8*y, (3, 1, 1), 1)
===

In [86]:

zt, zm, dic = zetas(n, pols, info=True)

In [90]:

quo, L = zt.partial_fraction_decomposition()
print(quo)
for p in L:
    print(p.factor())

0
(1737/350) * (s + 9/14)^-1
(-56/15) * (s + 2/3)^-1
(-28/25) * (s + 1)^-1

In [94]:

for j in dic:
    print("El ciclotómico de orden ", j, " tiene multiplicidad ", dic[j])

El ciclotómico de orden  1  tiene multiplicidad  3
El ciclotómico de orden  2  tiene multiplicidad  5
El ciclotómico de orden  3  tiene multiplicidad  3
El ciclotómico de orden  5  tiene multiplicidad  2
El ciclotómico de orden  14  tiene multiplicidad  5
El ciclotómico de orden  15  tiene multiplicidad  2

In [108]:

n1a, Nf1a, Ng1a, Nd1a, Nhs1a, Nks1a, euler1a, nus1a, pols1a = trinomiales_general(A, Vf, Vg, Vhs, Vks, Vd, total=False)

In [110]:

n1 == n1a and Nf1 == Nf1a and Ng1 == Ng1a and nus1 == nus1a and euler1 == euler1a

Out[110]:

True

In [111]:

Nhs1

Out[111]:

[((1, 2, 3, 1, 2), 1), ((1, 1, 2, 2, 3), 1)]

In [112]:

Nhs1a

Out[112]:

[((2, 3, 5, 3, 5), 1)]

In [113]:

pols1a.keys()

Out[113]:

dict_keys([0, 1, 2, 3, 4, (0, 2), (0, 4), (1, 2), (3, 4), (0, 'd'), (2, 'd'), (4, 'd'), (1, ('h', 0)), (3, ('h', 0))])

In [ ]: