Steenbrink/formastrinomialesysusresiduos.ipynb

In [4]:

S.<m3, nu1, g3, s> = PolynomialRing(QQ, order='neglex')

Primer caso: $f=z^3 - x^{m_3}$, $\omega=x^{\nu_1-1} d x d z$ y $g_3=\gcd(3,m_3)$.

In [17]:

tau1 = vector([m3, 0])
tau0 = vector(S, [0, 3])
V1 = vector(S, [0, 1])
V0 = vector(S, [1, 0])

In [39]:

Vgamma = vector([3, m3]) / g3
omega1 = vector([nu1, 1])
mu1 = 3 / g3
Pgamma = (tau0 * s + omega1) * Vgamma
volgamma = g3
P1 = (tau1 * s + omega1) * V1
P0 = (tau0 * s + omega1) * V0
Pgamma, P1, P0

Out[39]:

((3*nu1 + m3 + 3*m3*s)/g3, 1, nu1)

In [32]:

mu1 = Matrix([Vgamma, V1]).det()
mu0 = Matrix([V0, Vgamma]).det()
mu1, mu0

Out[32]:

(3/g3, m3/g3)

In [38]:

Jtau1 = mu1 / P1 / Pgamma
Jtau0 = mu0 / P0 / Pgamma
Jgamma = 1 / Pgamma

Out[38]:

g3/(3*nu1 + m3 + 3*m3*s)

In [41]:

Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma
Zeta.factor()

Out[41]:

(-1) * nu1^-1 * (1 + s)^-1 * (3*nu1 + m3 + 3*m3*s)^-1 * (-3*nu1 - 3*nu1*s + nu1*g3^2*s - m3 - m3*s)

In [49]:

R1 = (Zeta * (s + 1))(s=-1)

In [57]:

Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))

In [58]:

Zeta - R1 / (s + 1) - Rp / Pgamma

Out[58]:

0

In [64]:

Zeta2 = Zeta(nu1=2 * m3 / 3)
Zeta2.factor()

Out[64]:

(-1/6) * (1 + s)^-2 * m3^-1 * (-9 - 9*s + 2*g3^2*s)

In [76]:

R2a = (Zeta2 * (s + 1)^2)(s=-1)
R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)
R1a.factor()

Out[76]:

(-1/6) * m3^-1 * (-9 + 2*g3^2)

In [77]:

Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2

Out[77]:

0

Segundo caso: $f=z^3 + x^{m_2}z + x^{m_3}$

In [79]:

S.<m2, nu1, g2, s> = PolynomialRing(QQ, order='neglex')

In [80]:

tau1 = vector([m2, 1])
tau0 = vector(S, [0, 3])
V1 = vector(S, [0, 1])
V0 = vector(S, [1, 0])

In [86]:

Vgamma = vector([2, m2]) / g2
omega1 = vector([nu1, 1])
mu1 = Matrix([Vgamma, V1]).det()
mu0 = Matrix([V0, Vgamma]).det()
Pgamma = (tau0 * s + omega1) * Vgamma
volgamma = g2
P1 = (tau1 * s + omega1) * V1
P0 = (tau0 * s + omega1) * V0

In [87]:

Jtau1 = mu1 / P1 / Pgamma
Jtau0 = mu0 / P0 / Pgamma
Jgamma = 1 / Pgamma

In [89]:

Jgamma.factor()

Out[89]:

g2 * (2*nu1 + m2 + 3*m2*s)^-1

In [91]:

Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma
show(Zeta.factor())

<html>\(\displaystyle \left(-1\right) \cdot \nu_{1}^{-1} \cdot (1 + s)^{-1} \cdot (2 \nu_{1} + m_{2} + 3 m_{2} s)^{-1} \cdot (-2 \nu_{1} + \nu_{1} g_{2}^{2} s - m_{2} - m_{2} s)\)</html>

In [93]:

R1 = (Zeta * (s + 1))(s=-1)
Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))
Rp.factor()

Out[93]:

(1/2) * (-nu1 + m2)^-1 * nu1^-1 * g2^-1 * (2*nu1 + m2) * (nu1*g2^2 + 2*m2)

In [94]:

Zeta - R1 / (s + 1) - Rp / Pgamma

Out[94]:

0

In [95]:

Pgamma(s=-1)

Out[95]:

(2*nu1 - 2*m2)/g2

In [96]:

Zeta2 = Zeta(nu1=m2)
Zeta2.factor()

Out[96]:

(-1/3) * (1 + s)^-2 * m2^-1 * (-3 - s + g2^2*s)

In [99]:

R2a = (Zeta2 * (s + 1)^2)(s=-1)
R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)
R1a.factor(), R2a.factor()

Out[99]:

((-1/3) * m2^-1 * (-1 + g2) * (1 + g2), (1/3) * m2^-1 * (2 + g2^2))

In [98]:

Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2

Out[98]:

0

Tercer caso: $f=z^3 + x^{m_1}z^2 + x^{m_3}$, $m_3 > 3 m_1$.

In [100]:

S.<m1, m3, nu1, g2, s> = PolynomialRing(QQ, order='neglex')

In [11]:

Tau2 - Gamma2*s / (s+1) + Tau1 - d*Gamma1*s / (s+1) + Tau0

Out[11]:

(2*c - c*d^2*s + a + a*s)/(2*c^2 + 2*c^2*s + a*c + 4*a*c*s + 3*a*c*s^2)

In [101]:

tau2 = vector([m3, 0])
tau1 = vector([m1, 2])
tau0 = vector(S, [0, 3])
V2 = vector(S, [0, 1])
V0 = vector(S, [1, 0])

In [ ]:

Vgamma1 = vector([1, m1]) 
Vgamma2 = vector([2, m3 - m1]) / g2

In [ ]: