Steenbrink/DeformacionGenerica.ipynb

In [1]:

# load('zetas_triples_general.py')
R.<x, y, z> = PolynomialRing(QQ, order='neglex')
S.<u, v> = PolynomialRing(QQ, order='neglex')
f = (v - u^2)^2 + u * v^3
g = f^2 + v^6
p = v - u^2
F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)
h = F.discriminant(z)(x=u, y=v)

In [2]:

F.jacobian_ideal().vector_space_dimension()

Out[2]:

41

In [4]:

# (F + 1 * p(u=x, v=y) * z^2).jacobian_ideal().vector_space_dimension()

In [43]:

# load('zetas_triples_general.py')
R.<x, y, z> = PolynomialRing(QQ, order='neglex')
S.<u, v> = PolynomialRing(QQ, order='neglex')
f = v^2 - u^7 + u^5 * v
f2 = v^2
g = f^2 + (v + u^2)^6
g1 = 3 / 2 * v * f
g3 = v^3
F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)
h = F.discriminant(z)(x=u, y=v)

In [44]:

F.jacobian_ideal().vector_space_dimension()

Out[44]:

41

In [53]:

t = -3
F1 = F - 3 * f2(u=x, v=y) * t^2 * z + 2 * g1(u=x, v=y) * t +  2 * g3(u=x, v=y) * t^3

In [54]:

F1.jacobian_ideal().vector_space_dimension()

Out[54]:

41

In [47]:

def nwt(p):
    return p(u=u^2, v=u^7 * (v + 1))

In [48]:

nwt(f).factor()

Out[48]:

u^14 * (2*v + v^2 + u^3 + u^3*v)

In [49]:

nwt(g).factor()

Out[49]:

u^24 * (1 + 6*u^3 + 6*u^3*v + 4*u^4*v^2 + 4*u^4*v^3 + u^4*v^4 + 15*u^6 + 30*u^6*v + 15*u^6*v^2 + 4*u^7*v + 6*u^7*v^2 + 2*u^7*v^3 + 20*u^9 + 60*u^9*v + 60*u^9*v^2 + 20*u^9*v^3 + u^10 + 2*u^10*v + u^10*v^2 + 15*u^12 + 60*u^12*v + 90*u^12*v^2 + 60*u^12*v^3 + 15*u^12*v^4 + 6*u^15 + 30*u^15*v + 60*u^15*v^2 + 60*u^15*v^3 + 30*u^15*v^4 + 6*u^15*v^5 + u^18 + 6*u^18*v + 15*u^18*v^2 + 20*u^18*v^3 + 15*u^18*v^4 + 6*u^18*v^5 + u^18*v^6)

In [50]:

nwt(f2).factor()

Out[50]:

(1 + v)^2 * u^14

In [39]:

nwt(g1).factor()

Out[39]:

(3/2) * (1 + v) * u^21 * (2*v + v^2 + u^3 + u^3*v)

In [40]:

nwt(g3).factor()

Out[40]:

(1 + v)^3 * u^21

In [41]:

h1 = F1.discriminant(z)(x=u, y=v)

In [42]:

Z1 = ZetaFunctions(h1)
[q.factor() for q in Z1.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]

Out[42]:

[(-27) * (v^2 - u^7) * (3965*v^4 + 2735*u^7*v^2 + 2048*u^14)]

In [20]:

h1a = S(h1(u=u^2, v=u^7 * (1 + v)) / u^21)
Z1a = ZetaFunctions(h1a)
[q.factor() for q in Z1a.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]

Out[20]:

[(108) * u^21 * (3*v^2 - 2*u^3)]

In [55]:

Rt.<x, y, z, t> = QQ[]
Ft = Rt(F - 3 * f2(u=x, v=y) * t^2 * z + 2 * g1(u=x, v=y) * t +  2 * g3(u=x, v=y) * t^3)

In [58]:

J = Rt.ideal([Ft.derivative(a) for a in Rt.gens()[:-1]] + [Ft])

In [66]:

Gt = Ft(z=z + y * t)

In [67]:

Gt.derivative(t)

Out[67]:

3*y*z^2

In [ ]: