Steenbrink/Untitled.ipynb

In [47]:

load('../Descargas/ZetaFunctionsNewtonND.py')
R.<x,y,z> = QQ[]
S.<u,v> = QQ[]
T.<t> = QQ[]
F = z^3 + 3 * (y^2 - x^3 + y^3) * z^2 + 3 * ((y^2 - x^3)^2 + x^2 * y^3 + x^7) * z + ((y^2 - x^3)^3 + x^2 * y^5) 

In [2]:

Z = ZetaFunctions(F)

In [3]:

Z.is_newton_degenerated()

Out[3]:

True

In [4]:

F1 = F(z=z - y^2 + x^3 - y^3)
F1.coefficient({z:2})

Out[4]:

0

In [6]:

Z = ZetaFunctions(F1)
# Z.is_newton_degenerated()

In [7]:

f1, f2 = [F1.coefficient({z: j})(x=u, y=v) for j in (1, 0)]

In [8]:

f1.newton_polytope()

Out[8]:

In [11]:

Z1 = ZetaFunctions(f1)
Z1.is_newton_degenerated()

Out[11]:

False

In [12]:

f2.newton_polytope()

Out[12]:

In [13]:

Z2 = ZetaFunctions(f2)
Z2.is_newton_degenerated()

Out[13]:

False

In [18]:

Z.give_info_newton(compact=True)

Newton's polyhedron of 3*x^10 - 3*x^7*y^3 - 3*x^7*y^2 + 3*x^6*y^3 - 6*x^3*y^6 + 2*y^9 + 3*x^5*y^3 - 6*x^3*y^5 - 3*x^2*y^6 + 6*y^8 + 3*x^7*z - 2*x^2*y^5 + 3*y^7 + 6*x^3*y^3*z - 3*y^6*z + 3*x^2*y^3*z - 6*y^5*z + z^3:
    support points = [(10, 0, 0), (7, 3, 0), (7, 2, 0), (6, 3, 0), (3, 6, 0), (0, 9, 0), (5, 3, 0), (3, 5, 0), (2, 6, 0), (0, 8, 0), (7, 0, 1), (2, 5, 0), (0, 7, 0), (3, 3, 1), (0, 6, 1), (2, 3, 1), (0, 5, 1), (0, 0, 3)]
    vertices = [(0, 0, 3), (10, 0, 0), (0, 7, 0), (5, 3, 0), (2, 5, 0)]
    number of compact faces = 15
    Facet 0: 3*x + 5*y + 10*z - 30 >= 0
    Facet 1: 3*x + 3*y + 7*z - 21 >= 0
    Facet 3: 6*x + 9*y + 19*z - 57 >= 0

In [15]:

J = F.jacobian_ideal() + R.ideal(F)
J.minimal_associated_primes()

Out[15]:

[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]

In [32]:

A = list(Z.get_polyfaces_dictionary(compact=True).keys())
A = [p for p in A if p.number_of_terms() > 2]
[p.jacobian_ideal().minimal_associated_primes() for p in A]

Out[32]:

[[Ideal (z, x) of Multivariate Polynomial Ring in x, y, z over Rational Field],
 [Ideal (z, y) of Multivariate Polynomial Ring in x, y, z over Rational Field],
 [Ideal (z, y) of Multivariate Polynomial Ring in x, y, z over Rational Field,
  Ideal (z, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]]

In [48]:

F = z^3 + 3 * ((y^2 - x^3)^2 + x^2 * y^3 ) * z + 2 * ((y^2 - x^3)^3 + x^2 * y^5) 

In [49]:

J = F.jacobian_ideal() + R.ideal(F)
J.minimal_associated_primes()

Out[49]:

[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]

In [50]:

Z = ZetaFunctions(F)
Z.is_newton_degenerated(local=True)

Out[50]:

True

In [92]:

A = list(Z.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[92]:

[-2*x^9 + 3*x^6*z + z^3,
 2*y^6 + 3*y^4*z + z^3,
 (2) * (-x^3 + y^2)^3,
 -2*x^9 + 6*x^6*y^2 - 6*x^3*y^4 + 3*x^6*z + 2*y^6 - 6*x^3*y^2*z + 3*y^4*z + z^3]

In [51]:

Z.give_info_newton(compact=True)

Newton's polyhedron of -2*x^9 + 6*x^6*y^2 - 6*x^3*y^4 + 2*x^2*y^5 + 3*x^6*z + 2*y^6 - 6*x^3*y^2*z + 3*x^2*y^3*z + 3*y^4*z + z^3:
    support points = [(9, 0, 0), (6, 2, 0), (3, 4, 0), (2, 5, 0), (6, 0, 1), (0, 6, 0), (3, 2, 1), (2, 3, 1), (0, 4, 1), (0, 0, 3)]
    vertices = [(0, 0, 3), (9, 0, 0), (0, 6, 0)]
    number of compact faces = 7
    Facet 3: 2*x + 3*y + 6*z - 18 >= 0

In [52]:

dis = F.discriminant(z)(x=u, y=v)

In [54]:

sum(dis.monomial_coefficient(u^(3 * j) * v^(2 * (6 - j))) * t^j for j in (0 .. 6)).factor()

Out[54]:

(-216) * (t - 1)^6

In [59]:

dis1 = S(dis(u=(u + 1) * v^2, v=(u + 1) * v^3) / v^36 / (u + 1)^12)
dis1.newton_polytope()

Out[59]:

In [60]:

sum(dis1.monomial_coefficient(u^(3 * j) * v^(1 * (2 - j))) * t^j for j in (0 .. 2)).factor()

Out[60]:

(-216) * (t^2 - t + 1/2)

In [70]:

G1 = F(x=x * y)

In [73]:

[G1.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[73]:

[1,
 (3) * v^4 * (u^6*v^2 - 2*u^3*v + u^2*v + 1),
 (-2) * v^6 * (u^9*v^3 - 3*u^6*v^2 + 3*u^3*v - u^2*v - 1)]

In [78]:

(t-1)^2*(t+2), (t+1)^2*(t-2)

Out[78]:

(t^3 - 3*t + 2, t^3 - 3*t - 2)

In [76]:

ZG1 = ZetaFunctions(G1)
ZG1.is_newton_degenerated()

Out[76]:

False

In [77]:

G2 = F(y=x^2 * y)
[G2.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[77]:

[1,
 (3) * u^6 * (u^2*v^4 + u^2*v^3 - 2*u*v^2 + 1),
 (2) * u^9 * (u^3*v^6 + u^3*v^5 - 3*u^2*v^4 + 3*u*v^2 - 1)]

In [81]:

G2a = F(y=x * y)(x=x * y)

In [82]:

G3a = G2a(x=x * y)
[G3a.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[82]:

[1,
 (3) * u^4 * v^12 * (u^2 + u*v - 2*u + 1),
 (2) * u^6 * v^18 * (-u^3 + 3*u^2 + u*v - 3*u + 1)]

In [84]:

G3b = G2a(y=x * y)
[G3b.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[84]:

[1,
 (3) * v^6 * u^12 * (u*v^2 + v^2 - 2*v + 1),
 (2) * v^9 * u^18 * (u*v^3 + v^3 - 3*v^2 + 3*v - 1)]

In [95]:

G3c = G2a(x=(x + 1) * y)
[G3c.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[95]:

[1,
 (3) * (u + 1)^4 * v^12 * (u^2 + u*v + v),
 (2) * (u + 1)^6 * v^18 * (-u^3 + u*v + v)]

In [96]:

G4 = G3c(x=x * y)
[G4.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[96]:

[1,
 (3) * v^13 * (u*v + 1)^4 * (u^2*v + u*v + 1),
 (-2) * v^19 * (u*v + 1)^6 * (u^3*v^2 - u*v - 1)]

In [97]:

G5 = G3c(y=x * y)(x=x * y)
[G5.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[97]:

[1,
 (3) * u^13 * v^26 * (u*v + u + 1) * (u*v + 1)^4,
 (-2) * u^19 * v^38 * (u*v + 1)^6 * (u^2*v - u*v - 1)]

In [98]:

G6 = G3c(y=x^3 * y)
[G6.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[98]:

[1,
 (3) * (u + 1)^4 * v^12 * u^38 * (u^2*v + u*v + 1),
 (2) * (u + 1)^6 * v^18 * u^57 * (u*v + v - 1)]

In [99]:

G7 = G6(y=y + 1)
[G7.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[99]:

[1,
 (3) * (u + 1)^4 * (v + 1)^12 * u^38 * (u^2*v + u^2 + u*v + u + 1),
 (2) * (u + 1)^6 * (v + 1)^18 * u^57 * (u*v + u + v)]

In [ ]:

In [ ]:

In [102]:

F = z^3 - 3 * ((y^2 - x^3)^2 + x^2 * y^3 + x^7 + y^5 ) * z + 2 * ((y^2 - x^3)^3 + x^2 * y^5) 

In [103]:

J = F.jacobian_ideal() + R.ideal(F)
J.minimal_associated_primes()

Out[103]:

[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]

In [104]:

Z = ZetaFunctions(F)
Z.is_newton_degenerated(local=True)

Out[104]:

True

In [105]:

A = list(Z.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[105]:

[(-2*x^3 + z) * (x^3 + z)^2,
 (2*y^2 + z) * (-y^2 + z)^2,
 (2) * (-x^3 + y^2)^3,
 (-2*x^3 + 2*y^2 + z) * (x^3 - y^2 + z)^2]

In [106]:

Z.give_info_newton(compact=True)

Newton's polyhedron of -2*x^9 + 6*x^6*y^2 - 3*x^7*z - 6*x^3*y^4 + 2*x^2*y^5 - 3*x^6*z + 2*y^6 + 6*x^3*y^2*z - 3*x^2*y^3*z - 3*y^5*z - 3*y^4*z + z^3:
    support points = [(9, 0, 0), (6, 2, 0), (7, 0, 1), (3, 4, 0), (2, 5, 0), (6, 0, 1), (0, 6, 0), (3, 2, 1), (2, 3, 1), (0, 5, 1), (0, 4, 1), (0, 0, 3)]
    vertices = [(0, 0, 3), (9, 0, 0), (0, 6, 0)]
    number of compact faces = 7
    Facet 1: 2*x + 3*y + 6*z - 18 >= 0

In [109]:

G1 = F(x=x * y)

In [123]:

ZG1 = ZetaFunctions(G1)
A = list(ZG1.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[123]:

[(2*y^2 + z) * (-y^2 + z)^2]

In [124]:

[G1.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[124]:

[1,
 (-3) * v^4 * (u^7*v^3 + u^6*v^2 - 2*u^3*v + u^2*v + v + 1),
 (-2) * v^6 * (u^9*v^3 - 3*u^6*v^2 + 3*u^3*v - u^2*v - 1)]

In [125]:

H1 = G1(z=z + y^2) 

In [126]:

ZH1 = ZetaFunctions(H1)
# ZH1.is_newton_degenerated()

In [127]:

A = list(ZH1.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[127]:

[]

In [129]:

ZH1.give_info_newton(compact=True)

Newton's polyhedron of -2*x^9*y^9 - 3*x^7*y^9 - 3*x^7*y^7*z + 3*x^6*y^8 - 3*x^6*y^6*z - x^2*y^7 + 6*x^3*y^5*z - 3*x^2*y^5*z - 3*y^7 - 3*y^5*z + 3*y^2*z^2 + z^3:
    support points = [(9, 9, 0), (7, 9, 0), (7, 7, 1), (6, 8, 0), (6, 6, 1), (2, 7, 0), (3, 5, 1), (2, 5, 1), (0, 7, 0), (0, 5, 1), (0, 2, 2), (0, 0, 3)]
    vertices = [(0, 0, 3), (0, 2, 2), (0, 7, 0)]
    number of compact faces = 5

In [130]:

G1a = F(y=x * y)

In [132]:

G2 = G1a(y=x * y)
[G2.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[132]:

[1,
 (-3) * u^6 * (u^4*v^5 + u^2*v^4 + u^2*v^3 - 2*u*v^2 + u + 1),
 (2) * u^9 * (u^3*v^6 + u^3*v^5 - 3*u^2*v^4 + 3*u*v^2 - 1)]

In [133]:

H2 = G2(z=z - x^3) 
ZH2 = ZetaFunctions(H2)
A = list(ZH2.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[133]:

[]

In [134]:

G2a = G1a(x=x * y)

In [141]:

G3 = G2a(x=x * y)
[G3.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[141]:

[1,
 (-3) * u^4 * v^12 * (u^3*v^2 + u*v^3 + u^2 + u*v - 2*u + 1),
 (2) * u^6 * v^18 * (-u^3 + 3*u^2 + u*v - 3*u + 1)]

In [154]:

H3 = G3(z=z + x^2 * y^6 - x^3 * y^6) 

In [155]:

H3.monomial_coefficient(z^2 * x^2 * y^6)

Out[155]:

3

In [156]:

ZH3 = ZetaFunctions(H3)
A = list(ZH3.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[156]:

[]

In [138]:

G4 = G2a(y=x * y)
[G4.coefficient({z: j})(x=u, y=v).factor() for j in (3, 1, 0)]

Out[138]:

[1,
 (-3) * v^6 * u^12 * (u^3*v^4 + u^2*v + u*v^2 + v^2 - 2*v + 1),
 (2) * v^9 * u^18 * (u*v^3 + v^3 - 3*v^2 + 3*v - 1)]

In [160]:

H4 = G4(z=z - x^6 * y^3 + x^6 * y^4) 
ZH4 = ZetaFunctions(H4)
A = list(ZH4.get_polyfaces_dictionary(compact=True).keys())
[p.factor() for p in A if p.number_of_terms() > 2]

Out[160]:

[(-3) * y^3 * x^6 * (-x^14*y^7 - x^13*y^8 + z^2)]

In [ ]: