100 KiB
100 KiB
In [1]:
# load('zetas_triples_general.py') R.<x, y, z> = PolynomialRing(QQ, order='neglex') S.<u, v> = PolynomialRing(QQ, order='neglex') f = v^2 - u^2 g = u^4 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]:
10
In [3]:
R0.<t> = QQ[] Kt = R0.fraction_field() Rt.<x, y, z> = PolynomialRing(Kt, order='neglex') S.<u, v> = PolynomialRing(Kt, order='neglex') f = v^2 - u^2 g = u^4 F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y) h = F.discriminant(z)(x=u, y=v) F.jacobian_ideal().vector_space_dimension()
Out[3]:
10
In [4]:
Ft = F + t * x * z^2 Ft.jacobian_ideal().vector_space_dimension()
Out[4]:
10
In [5]:
load('zetas_triples_general.py') ht = Ft.discriminant(z)(x=u, y=v)
In [8]:
Zt = ZetaFunctions(ht) Z = ZetaFunctions(h)
In [11]:
[p.factor() for p in Z.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[11]:
[(-108) * (-v + u)^3 * (v + u)^3]
In [12]:
[p.factor() for p in Zt.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[12]:
[(9) * (v - u)^2 * (v + u)^2 * (12*v^2 + (t^2 - 12)*u^2)]
In [13]:
Zta = ZetaFunctions(ht(v=v + u)) Za = ZetaFunctions(h(v=v + u))
In [15]:
[p.factor() for p in Za.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[15]:
[(108) * v^3 * (v + 2*u)^3, (108) * u^3 * (8*v^3 - u^5)]
In [16]:
[p.factor() for p in Zta.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[16]:
[(9) * v^2 * (v + 2*u)^2 * (12*v^2 + 24*u*v + (t^2)*u^2), ((4*t^2)) * u^4 * (9*v^2 + (-2*t)*u^3)]
In [17]:
Ztb = ZetaFunctions(ht(v=v - u)) Zb = ZetaFunctions(h(v=v -u ))
In [18]:
[p.factor() for p in Zb.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[18]:
[(-108) * (-v + 2*u)^3 * v^3, (-108) * u^3 * (8*v^3 + u^5)]
In [19]:
[p.factor() for p in Ztb.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]
Out[19]:
[(9) * v^2 * (v - 2*u)^2 * (12*v^2 - 24*u*v + (t^2)*u^2), ((4*t^2)) * u^4 * (9*v^2 + (-2*t)*u^3)]
In [24]:
Zt = ZetaFunctions(Ft) Z = ZetaFunctions(F)
In [47]:
def nwt(p): return p(u=u^2, v=u^7 * (v + 1))
In [27]:
Z.newton_plot()
Out[27]:
In [28]:
Zt.newton_plot()
Out[28]:
In [30]:
Z.give_info_newton(compact=True)
Newton's polyhedron of z^3 - 3*y^2*z + 3*x^2*z + 2*x^4:
support points = [(0, 0, 3), (0, 2, 1), (2, 0, 1), (4, 0, 0)]
vertices = [(0, 0, 3), (0, 2, 1), (2, 0, 1), (4, 0, 0)]
number of compact faces = 11
Facet 1: -4 + 2*z + y + x >= 0
Facet 4: -3 + z + y + x >= 0
In [31]:
Zt.give_info_newton(compact=True)
Newton's polyhedron of z^3 - 3*y^2*z + t*x*z^2 + 3*x^2*z + 2*x^4:
support points = [(0, 0, 3), (0, 2, 1), (1, 0, 2), (2, 0, 1), (4, 0, 0)]
vertices = [(0, 0, 3), (0, 2, 1), (4, 0, 0), (2, 0, 1)]
number of compact faces = 11
Facet 0: -4 + 2*z + y + x >= 0
Facet 4: -3 + z + y + x >= 0
In [32]:
Ft
Out[32]:
z^3 - 3*y^2*z + t*x*z^2 + 3*x^2*z + 2*x^4
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 [ ]: