Steenbrink/UnLado.ipynb

In [113]:

R.<ss, a1, a2, alpha1, alpha2, alpha, nu1, nu2, t, p1, p2, cd, rt> = QQ[]

In [115]:

e1, e2, e3 = [v.change_ring(R) for v in (QQ^3).gens()]

In [116]:

V1 = vector([0, 0, 3])
V0 = vector([a1, a2, 0])
nu = vector([nu1, nu2, 1])

In [117]:

Q1 = vector([3, 0, a1]) / alpha1
Q2 = vector([0, 3, a2]) / alpha2

In [118]:

def cono3(u, v, w, vert, nu0):
    m = Matrix([u, v, w]).det()
    den = 1
    for u0 in (u, v, w):
        den *= (u0 * vert) * ss + u0 * nu0
    return m / den

In [119]:

sb = {ss: t - 1 / 3}

In [120]:

vertical = cono3(e1, e2, e3, V1, nu) - cono3(Q1, Q2, e3, V1, nu)
show(vertical.subs(sb).factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1} \cdot (a_{1} a_{2} t + a_{2} \nu_{1} + a_{1} \nu_{2})\)</html>

In [121]:

horizontal = cono3(Q1, Q2, e3, V0, nu)
horizontal = (horizontal.numerator() / 9) / (horizontal.denominator() / 9)
show(horizontal.subs(sb).factor())

<html>\(\displaystyle (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1}\)</html>

In [122]:

show((vertical + horizontal).subs(sb).factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1} \cdot (a_{1} a_{2} t + a_{2} \nu_{1} + a_{1} \nu_{2} + 3 \nu_{1} \nu_{2})\)</html>

In [123]:

Matrix([Q1, Q2]).minors(2)

Out[123]:

[9/(alpha1*alpha2), 3*a2/(alpha1*alpha2), (-3*a1)/(alpha1*alpha2)]

In [124]:

mgamma = 3 * cd / alpha1 / alpha2

In [125]:

Q1 * V1, Q1 * V0, Q2 * V1, Q2 * V0, Q1 * nu

Out[125]:

(3*a1/alpha1, 3*a1/alpha1, 3*a2/alpha2, 3*a2/alpha2, (a1 + 3*nu1)/alpha1)

In [126]:

f1 = ((Q1 * V1) * ss + Q1 * nu)
f2 = ((Q2 * V1) * ss + Q2 * nu)

In [127]:

arista = mgamma * ss / (ss + 1) / f1 / f2 * cd 
show(arista.subs(sb).factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot (3 t + 2)^{-1} \cdot (3 t - 1) \cdot \mathit{cd}^{2} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1}\)</html>

In [128]:

prueba = horizontal - arista + vertical
# prueba.numerator().variables() + prueba.denominator().variables()
show(prueba.subs(sb).factor())

<html>\(\displaystyle \left(-\frac{1}{3}\right) \cdot (3 t + 2)^{-1} \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1} \cdot (3 \nu_{1} \nu_{2} t \mathit{cd}^{2} - 3 a_{1} a_{2} t^{2} - \nu_{1} \nu_{2} \mathit{cd}^{2} - 2 a_{1} a_{2} t - 3 a_{2} \nu_{1} t - 3 a_{1} \nu_{2} t - 9 \nu_{1} \nu_{2} t - 2 a_{2} \nu_{1} - 2 a_{1} \nu_{2} - 6 \nu_{1} \nu_{2})\)</html>

In [129]:

def compara(aa1, aa2, nnu1, nnu2):
    R.<x, y, z> = QQ[]
    F = z^3 + x^aa1 * y^aa2
    zt = ZetaFunctions(F).topological_zeta(local=True, weights=(nnu1, nnu2, 1))
    S = zt.parent()
    s = S.gen(0)
    aalpha1 = aa1.gcd(3)
    aalpha2 = aa2.gcd(3)
    ccd = aalpha1.gcd(aalpha2)
    pr = prueba(nu1=nnu1, nu2=nnu2, a1=aa1, a2=aa2, cd=ccd, ss=s)
    return (zt, pr)

In [130]:

zt, pr = compara(5, 7, 2, 3)

In [131]:

zt.factor()

Out[131]:

(1/18) * (s + 11/15)^-1 * (s + 16/21)^-1 * (s + 1)^-1 * (s^2 + 263/105*s + 176/105)

In [132]:

pr.factor()

Out[132]:

(1/18) * (s + 11/15)^-1 * (s + 16/21)^-1 * (s + 1)^-1 * (s^2 + 263/105*s + 176/105)

In [133]:

zt - pr

Out[133]:

0

In [134]:

zt, pr = compara(6, 9, 2, 5)

In [135]:

zt.factor()

Out[135]:

(1/30) * (s + 2/3)^-1 * (s + 8/9)^-1 * (s + 1)^-1 * (s^2 + 10/9*s + 16/9)

In [136]:

pr.factor()

Out[136]:

(1/30) * (s + 2/3)^-1 * (s + 8/9)^-1 * (s + 1)^-1 * (s^2 + 10/9*s + 16/9)

In [137]:

zt - pr

Out[137]:

0

In [138]:

zt, pr = compara(5, 0, 2, 1)

In [139]:

zt.factor()

Out[139]:

(3/10) * (s + 11/15)^-1 * (s + 1)^-1 * (s + 11/9)

In [140]:

pr.factor()

Out[140]:

(3/10) * (s + 11/15)^-1 * (s + 1)^-1 * (s + 11/9)

In [141]:

zt - pr

Out[141]:

0

In [142]:

zt, pr = compara(6, 0, 2, 1)

In [143]:

zt.factor()

Out[143]:

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

In [144]:

pr.factor()

Out[144]:

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

In [145]:

zt - pr

Out[145]:

0

In [146]:

prt = prueba.subs(sb)
show(prt.factor())

<html>\(\displaystyle \left(-\frac{1}{3}\right) \cdot (3 t + 2)^{-1} \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (a_{2} t + \nu_{2})^{-1} \cdot (a_{1} t + \nu_{1})^{-1} \cdot (3 \nu_{1} \nu_{2} t \mathit{cd}^{2} - 3 a_{1} a_{2} t^{2} - \nu_{1} \nu_{2} \mathit{cd}^{2} - 2 a_{1} a_{2} t - 3 a_{2} \nu_{1} t - 3 a_{1} \nu_{2} t - 9 \nu_{1} \nu_{2} t - 2 a_{2} \nu_{1} - 2 a_{1} \nu_{2} - 6 \nu_{1} \nu_{2})\)</html>

In [147]:

prt1 = prt * (a2 * t + nu2) * (a1 * t + nu1) * (3 * t + 2)
show(prt1.factor())

<html>\(\displaystyle \left(-\frac{1}{3}\right) \cdot \nu_{2}^{-1} \cdot \nu_{1}^{-1} \cdot (3 \nu_{1} \nu_{2} t \mathit{cd}^{2} - 3 a_{1} a_{2} t^{2} - \nu_{1} \nu_{2} \mathit{cd}^{2} - 2 a_{1} a_{2} t - 3 a_{2} \nu_{1} t - 3 a_{1} \nu_{2} t - 9 \nu_{1} \nu_{2} t - 2 a_{2} \nu_{1} - 2 a_{1} \nu_{2} - 6 \nu_{1} \nu_{2})\)</html>

In [148]:

cf23 = (prt1 / (a2 * t + nu2) / (a1 * t + nu1))(t=-2 / 3)
show(cf23.factor())

<html>\(\displaystyle \left(-9\right) \cdot (2 a_{2} - 3 \nu_{2})^{-1} \cdot (-2 a_{1} + 3 \nu_{1})^{-1} \cdot \mathit{cd}^{2}\)</html>

In [149]:

cf1 = (prt1 / (3 * t + 2) / (a2 * t + nu2))(t=-nu1 / a1)
show(cf1.factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \nu_{1}^{-1} \cdot (2 a_{1} - 3 \nu_{1})^{-1} \cdot a_{1} \cdot (a_{1} + 3 \nu_{1}) \cdot (-a_{2} \nu_{1} + a_{1} \nu_{2})^{-1} \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1} - 3 \nu_{1})\)</html>

In [150]:

cf2 = (prt1 / (3 * t + 2) / (a1 * t + nu1))(t=-nu2 / a2)
show(cf2.factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \nu_{2}^{-1} \cdot (2 a_{2} - 3 \nu_{2})^{-1} \cdot a_{2} \cdot (a_{2} + 3 \nu_{2}) \cdot (a_{2} \nu_{1} - a_{1} \nu_{2})^{-1} \cdot (\nu_{2} \mathit{cd}^{2} + 2 a_{2} - 3 \nu_{2})\)</html>

In [151]:

(cf23 / (3 * t + 2) + cf1 / (a1 * t + nu1) + cf2 / (a2 * t + nu2) - prt)

Out[151]:

0

In [152]:

cf23(a2=0, nu2=1) / (3 * t + 2) + cf1(a2=0, nu2=1) / (a1 * t + nu1) - prt(a2=0, nu2=1)

Out[152]:

0

In [153]:

show(cf23(a2=0, nu2=1).factor())

<html>\(\displaystyle \left(-3\right) \cdot (2 a_{1} - 3 \nu_{1})^{-1} \cdot \mathit{cd}^{2}\)</html>

In [154]:

show(cf1(a2=0, nu2=1).factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \nu_{1}^{-1} \cdot (2 a_{1} - 3 \nu_{1})^{-1} \cdot (a_{1} + 3 \nu_{1}) \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1} - 3 \nu_{1})\)</html>

In [155]:

prta = prt(a2=rt * a1, nu2=rt * nu1)
prta.factor()

Out[155]:

(-1/3) * nu1^-2 * rt^-1 * (3*t + 2)^-1 * (a1*t + nu1)^-2 * (3*nu1^2*t*cd^2 - 3*a1^2*t^2 - nu1^2*cd^2 - 2*a1^2*t - 6*a1*nu1*t - 9*nu1^2*t - 4*a1*nu1 - 6*nu1^2)

In [156]:

cf23a = cf23(a2=rt * a1, nu2=rt * nu1)
show(cf23a.factor())

<html>\(\displaystyle \left(9\right) \cdot (2 a_{1} - 3 \nu_{1})^{-2} \cdot \mathit{rt}^{-1} \cdot \mathit{cd}^{2}\)</html>

In [157]:

prtb = prta - cf23a / (3 * t + 2)

In [159]:

cf1a = (prtb * (a1 * t + nu1)^2)(t=-nu1 / a1)
show(cf1a.factor())
prtc = prtb - cf1a / (a1 * t + nu1)^2
cf2a = prtc * (a1 * t + nu1)
show(cf2a.factor())

<html>\(\displaystyle \left(\frac{1}{3}\right) \cdot \mathit{rt}^{-1} \cdot \nu_{1}^{-1} \cdot (2 a_{1} - 3 \nu_{1})^{-1} \cdot (a_{1} + 3 \nu_{1}) \cdot (\nu_{1} \mathit{cd}^{2} + 2 a_{1} - 3 \nu_{1})\)</html>

<html>\(\displaystyle \left(-\frac{1}{3}\right) \cdot \nu_{1}^{-2} \cdot (2 a_{1} - 3 \nu_{1})^{-2} \cdot \mathit{rt}^{-1} \cdot a_{1} \cdot (3 \nu_{1} \mathit{cd} - 2 a_{1} + 3 \nu_{1}) \cdot (3 \nu_{1} \mathit{cd} + 2 a_{1} - 3 \nu_{1})\)</html>

In [160]:

(cf23a / (3 * t + 2) + cf1a / (a1 * t + nu1)^2 + cf2a / (a1 * t + nu1) - prta)

Out[160]:

0

In [ ]: