Sage & Gröbner computations¶

In [1]:
P.<x,y> = PolynomialRing(QQ,order='deglex')
P
Out[1]:
Multivariate Polynomial Ring in x, y over Rational Field
In [2]:
p1 = 2*x^3-4*x*y
p2 = x^2*y-2*y^2+x
In [3]:
print(p1)
print(p2)

2*x^3 - 4*x*y
x^2*y - 2*y^2 + x
In [4]:
I = P.ideal(p1,p2)
I
Out[4]:
Ideal (2*x^3 - 4*x*y, x^2*y - 2*y^2 + x) of Multivariate Polynomial Ring in x, y over Rational Field
In [5]:
p1.lt()
Out[5]:
2*x^3
In [6]:
p1.lm()
Out[6]:
x^3
In [7]:
p1.lc()
Out[7]:
2
In [8]:
from sage.rings.polynomial.toy_buchberger import *
set_verbose(1)
G = buchberger(I)
G

(x^2*y - 2*y^2 + x, 2*x^3 - 4*x*y) => x^2
G: {x^2, 2*x^3 - 4*x*y, x^2*y - 2*y^2 + x}

(x^2*y - 2*y^2 + x, x^2) => -2*y^2 + x
G: {x^2, -2*y^2 + x, 2*x^3 - 4*x*y, x^2*y - 2*y^2 + x}

(2*x^3 - 4*x*y, x^2) => -2*x*y
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(-2*y^2 + x, -2*x*y) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(x^2*y - 2*y^2 + x, -2*x*y) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(x^2, -2*x*y) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(x^2, -2*y^2 + x) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(2*x^3 - 4*x*y, x^2*y - 2*y^2 + x) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(x^2*y - 2*y^2 + x, -2*y^2 + x) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(2*x^3 - 4*x*y, -2*x*y) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

(2*x^3 - 4*x*y, -2*y^2 + x) => 0
G: {-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x}

8 reductions to zero.
Out[8]:
[-2*y^2 + x, 2*x^3 - 4*x*y, x^2, -2*x*y, x^2*y - 2*y^2 + x]
In [9]:
inter_reduction?
In [10]:
inter_reduction(G)
Out[10]:
{y^2 - 1/2*x, x*y, x^2}
In [11]:
p3 = spol(p1,p2)
p3
Out[11]:
-x^2
In [12]:
xgamma = lcm(p1.lm(),p2.lm())
xgamma
Out[12]:
x^3*y
In [13]:
s = xgamma/p1.lt()*p1 - xgamma/p2.lt()*p2
s
Out[13]:
-x^2
In [14]:
s.parent()
Out[14]:
Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
In [15]:
s.lt()

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
Cell In [15], line 1
----> 1 s.lt()

File /home/local/hakaeero/sage/src/sage/structure/element.pyx:494, in sage.structure.element.Element.__getattr__()
    492         AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'
    493     """
--> 494     return self.getattr_from_category(name)
    495 
    496 cdef getattr_from_category(self, name):

File /home/local/hakaeero/sage/src/sage/structure/element.pyx:507, in sage.structure.element.Element.getattr_from_category()
    505     else:
    506         cls = P._abstract_element_class
--> 507     return getattr_from_other_class(self, cls, name)
    508 
    509 def __dir__(self):

File /home/local/hakaeero/sage/src/sage/cpython/getattr.pyx:361, in sage.cpython.getattr.getattr_from_other_class()
    359     dummy_error_message.cls = type(self)
    360     dummy_error_message.name = name
--> 361     raise AttributeError(dummy_error_message)
    362 attribute = <object>attr
    363 # Check for a descriptor (__get__ in Python)

AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'lt'
In [16]:
P(s).parent()
Out[16]:
Multivariate Polynomial Ring in x, y over Rational Field
In [17]:
P(s).lt()
Out[17]:
-x^2
In [18]:
I.groebner_basis()
Out[18]:
[x^2, x*y, y^2 - 1/2*x]

Example 7.23¶

In [19]:
P.<x,y,z> = PolynomialRing(QQ,order='deglex')
p1 = x*z-y^2
p2 = x^3-z^2
I = P.ideal(p1,p2)
G = buchberger(I)
G

(x*z - y^2, x^3 - z^2) => -x^2*y^2 + z^3
G: {x^3 - z^2, x*z - y^2, -x^2*y^2 + z^3}

(x^3 - z^2, x*z - y^2) => 0
G: {x^3 - z^2, x*z - y^2, -x^2*y^2 + z^3}

(x*z - y^2, -x^2*y^2 + z^3) => -x*y^4 + z^4
G: {x^3 - z^2, x*z - y^2, -x*y^4 + z^4, -x^2*y^2 + z^3}

(x^3 - z^2, -x^2*y^2 + z^3) => 0
G: {x^3 - z^2, x*z - y^2, -x*y^4 + z^4, -x^2*y^2 + z^3}

(-x^2*y^2 + z^3, -x*y^4 + z^4) => 0
G: {x^3 - z^2, x*z - y^2, -x*y^4 + z^4, -x^2*y^2 + z^3}

(x*z - y^2, -x*y^4 + z^4) => -y^6 + z^5
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

(x^3 - z^2, -x*y^4 + z^4) => 0
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

(-x*y^4 + z^4, -y^6 + z^5) => 0
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

(-x^2*y^2 + z^3, -y^6 + z^5) => 0
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

(x*z - y^2, -y^6 + z^5) => 0
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

(x^3 - z^2, -y^6 + z^5) => 0
G: {x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2}

8 reductions to zero.
Out[19]:
[x^3 - z^2, -x*y^4 + z^4, -y^6 + z^5, -x^2*y^2 + z^3, x*z - y^2]
In [20]:
f = -4*x^2*y^2*z^2+y^6+3*z^5
f
Out[20]:
-4*x^2*y^2*z^2 + y^6 + 3*z^5
In [21]:
f.reduce(G)
Out[21]:
0
In [22]:
for g in inter_reduction(G):
    print(g.lt())

x^3
x^2*y^2
x*z
x*y^4
y^6
In [23]:
f.lift(G)
Out[23]:
[0, 0, 3, 0, -4*x*y^2*z - 4*y^4]

Example 7.24¶

In [24]:
P.<l,x,y,z> = PolynomialRing(QQ,order='lex')
f = x^3+2*x*y*z-z^2
g = x^2+y^2+z^2-1
p1 = (f-l*g).derivative(x)
p2 = (f-l*g).derivative(y)
p3 = (f-l*g).derivative(z)
I = P.ideal(p1,p2,p3,g)
for h in I.gens():
    print(h)

-2*l*x + 3*x^2 + 2*y*z
-2*l*y + 2*x*z
-2*l*z + 2*x*y - 2*z
x^2 + y^2 + z^2 - 1
In [25]:
G = I.groebner_basis()
for i,gi in enumerate(G):
    print("g_%d = %s"%(i,gi))

g_0 = l - 3/2*x - 3/2*y*z - 167616/3835*z^6 + 36717/590*z^4 - 134419/7670*z^2
g_1 = x^2 + y^2 + z^2 - 1
g_2 = x*y - 19584/3835*z^5 + 1999/295*z^3 - 6403/3835*z
g_3 = x*z + y*z^2 - 1152/3835*z^5 - 108/295*z^3 + 2556/3835*z
g_4 = y^3 + y*z^2 - y - 9216/3835*z^5 + 906/295*z^3 - 2562/3835*z
g_5 = y^2*z - 6912/3835*z^5 + 827/295*z^3 - 3839/3835*z
g_6 = y*z^3 - y*z - 576/59*z^6 + 1605/118*z^4 - 453/118*z^2
g_7 = z^7 - 1763/1152*z^5 + 655/1152*z^3 - 11/288*z
In [26]:
set_verbose(0)
factor(G[7])
Out[26]:
(1/1152) * z * (z - 1) * (z + 1) * (3*z - 2) * (3*z + 2) * (128*z^2 - 11)
In [27]:
Pz.<z> = PolynomialRing(QQ)
h = Pz(G[-1])
h
Out[27]:
z^7 - 1763/1152*z^5 + 655/1152*z^3 - 11/288*z
In [28]:
QQbar
Out[28]:
Algebraic Field
In [29]:
roots = h.roots(QQbar)
roots
Out[29]:
[(-1.000000000000000?, 1),
 (-0.6666666666666667?, 1),
 (-0.2931509849889644?, 1),
 (0, 1),
 (0.2931509849889644?, 1),
 (0.6666666666666667?, 1),
 (1.000000000000000?, 1)]
In [30]:
for z0,mult in roots:
    z0.exactify()
roots
Out[30]:
[(-1, 1),
 (-2/3, 1),
 (-0.2931509849889644?, 1),
 (0, 1),
 (0.2931509849889644?, 1),
 (2/3, 1),
 (1, 1)]
In [31]:
for z0,mult in roots:
    print(z0.radical_expression())

-1
-2/3
-1/8*sqrt(11/2)
0
1/8*sqrt(11/2)
2/3
1
In [32]:
roots[2]
Out[32]:
(-0.2931509849889644?, 1)
In [33]:
b,mult = roots[2]
b^2
Out[33]:
11/128
In [34]:
for z0,mult in roots:
    print("\nz = %s"%z0.radical_expression())
    print("-"*30)
    for i,gi in enumerate(G):
        gis = gi(z=z0)
        if gis:
            for cc,mon in gis:
                cc.exactify()
        print("g_%d = %s"%(i,gis))

z = -1
------------------------------
g_0 = l + (-3/2)*x + 3/2*y + 1
g_1 = x^2 + y^2
g_2 = x*y
g_3 = -x + y
g_4 = y^3
g_5 = -y^2
g_6 = 0
g_7 = 0

z = -2/3
------------------------------
g_0 = l + (-3/2)*x + y + 2/3
g_1 = x^2 + y^2 - 5/9
g_2 = x*y - 2/9
g_3 = (-2/3)*x + 4/9*y - 8/27
g_4 = y^3 + (-5/9)*y - 4/27
g_5 = (-2/3)*y^2 + 2/27
g_6 = 10/27*y + 10/81
g_7 = 0

z = -1/8*sqrt(11/2)
------------------------------
g_0 = l + (-3/2)*x + 0.4397264774834466?*y - 275/256
g_1 = x^2 + y^2 - 117/128
g_2 = x*y + 0.3297948581125849?
g_3 = (-0.2931509849889644?)*x + 11/128*y - 0.1855096076883290?
g_4 = y^3 + (-117/128)*y + 0.1236730717922194?
g_5 = (-0.2931509849889644?)*y^2 + 0.2267339649524022?
g_6 = 0.2679583222164752?*y - 3861/16384
g_7 = 0

z = 0
------------------------------
g_0 = l + (-3/2)*x
g_1 = x^2 + y^2 - 1
g_2 = x*y
g_3 = 0
g_4 = y^3 - y
g_5 = 0
g_6 = 0
g_7 = 0

z = 1/8*sqrt(11/2)
------------------------------
g_0 = l + (-3/2)*x + (-0.4397264774834466?)*y - 275/256
g_1 = x^2 + y^2 - 117/128
g_2 = x*y - 0.3297948581125849?
g_3 = 0.2931509849889644?*x + 11/128*y + 0.1855096076883290?
g_4 = y^3 + (-117/128)*y - 0.1236730717922194?
g_5 = 0.2931509849889644?*y^2 - 0.2267339649524022?
g_6 = (-0.2679583222164752?)*y - 3861/16384
g_7 = 0

z = 2/3
------------------------------
g_0 = l + (-3/2)*x - y + 2/3
g_1 = x^2 + y^2 - 5/9
g_2 = x*y + 2/9
g_3 = 2/3*x + 4/9*y + 8/27
g_4 = y^3 + (-5/9)*y + 4/27
g_5 = 2/3*y^2 - 2/27
g_6 = (-10/27)*y + 10/81
g_7 = 0

z = 1
------------------------------
g_0 = l + (-3/2)*x + (-3/2)*y + 1
g_1 = x^2 + y^2
g_2 = x*y
g_3 = x + y
g_4 = y^3
g_5 = y^2
g_6 = 0
g_7 = 0
In [35]:
PA = P.change_ring(QQbar)
for z0,mult in roots:
    print("\nz = %s"%z0.radical_expression())
    print("-"*30)
    Gs = [gi(z=z0) for gi in G]
    for i,gi in enumerate(PA.ideal(Gs).groebner_basis()):
        print("g_%d = %s"%(i,gi))

z = -1
------------------------------
g_0 = l + 1
g_1 = x - y
g_2 = y^2

z = -2/3
------------------------------
g_0 = l + 4/3
g_1 = x + 2/3
g_2 = y + 1/3

z = -1/8*sqrt(11/2)
------------------------------
g_0 = l - 1/8
g_1 = x + 3/8
g_2 = y - 0.8794529549668931?

z = 0
------------------------------
g_0 = l - 3/2*x
g_1 = x^2 + y^2 - 1
g_2 = x*y
g_3 = y^3 - y

z = 1/8*sqrt(11/2)
------------------------------
g_0 = l - 1/8
g_1 = x + 3/8
g_2 = y + 0.8794529549668931?

z = 2/3
------------------------------
g_0 = l + 4/3
g_1 = x + 2/3
g_2 = y - 1/3

z = 1
------------------------------
g_0 = l + 1
g_1 = x + y
g_2 = y^2
In [36]:
solutions = solve([SR(gi) for gi in G],var('l x y z'),solution_dict=True)
solutions
Out[36]:
[{l: -4/3, x: -2/3, y: 1/3, z: 2/3},
 {l: -4/3, x: -2/3, y: -1/3, z: -2/3},
 {l: 1/8, x: -3/8, y: 3/16*sqrt(11)*sqrt(2), z: -1/16*sqrt(11)*sqrt(2)},
 {l: 1/8, x: -3/8, y: -3/16*sqrt(11)*sqrt(2), z: 1/16*sqrt(11)*sqrt(2)},
 {l: -1, x: 0, y: 0, z: -1},
 {l: -1, x: 0, y: 0, z: 1},
 {l: 0, x: 0, y: 1, z: 0},
 {l: 0, x: 0, y: -1, z: 0},
 {l: 3/2, x: 1, y: 0, z: 0},
 {l: -3/2, x: -1, y: 0, z: 0}]
In [37]:
for sol in solutions:
    print("f(%s,%s,%s,%s) = %s"%(*sol.values(),f(*sol.values())))

f(-4/3,-2/3,1/3,2/3) = -28/27
f(-4/3,-2/3,-1/3,-2/3) = -28/27
f(1/8,-3/8,3/16*sqrt(11)*sqrt(2),-1/16*sqrt(11)*sqrt(2)) = 7/128
f(1/8,-3/8,-3/16*sqrt(11)*sqrt(2),1/16*sqrt(11)*sqrt(2)) = 7/128
f(-1,0,0,-1) = -1
f(-1,0,0,1) = -1
f(0,0,1,0) = 0
f(0,0,-1,0) = 0
f(3/2,1,0,0) = 1
f(-3/2,-1,0,0) = -1

Example 7.25¶

In [38]:
from time import time
P.<x,y,z,w> = PolynomialRing(QQ,order='degrevlex')
for n in range(2,201):
    I = P.ideal(x^(n+1)-y*z^(n-1)*w, x*y^(n-1)-z^n,x^n*z-y^n*w)
    t1 = time()
    G = I.groebner_basis()
    t2 = time()
    print("n = %d: computation time %.5f seconds"%(n,t2-t1))
    print(G)

n = 2: computation time 0.00184 seconds
[z^5 - y^4*w, x*z^3 - y^3*w, x^3 - y*z*w, x^2*z - y^2*w, x*y - z^2]
n = 3: computation time 0.00118 seconds
[z^10 - y^9*w, x*z^7 - y^7*w, x^2*z^4 - y^5*w, x^4 - y*z^2*w, x^3*z - y^3*w, x*y^2 - z^3]
n = 4: computation time 0.00120 seconds
[z^17 - y^16*w, x*z^13 - y^13*w, x^2*z^9 - y^10*w, x^3*z^5 - y^7*w, x^5 - y*z^3*w, x^4*z - y^4*w, x*y^3 - z^4]
n = 5: computation time 0.00111 seconds
[z^26 - y^25*w, x*z^21 - y^21*w, x^2*z^16 - y^17*w, x^3*z^11 - y^13*w, x^4*z^6 - y^9*w, x^6 - y*z^4*w, x^5*z - y^5*w, x*y^4 - z^5]
n = 6: computation time 0.00108 seconds
[z^37 - y^36*w, x*z^31 - y^31*w, x^2*z^25 - y^26*w, x^3*z^19 - y^21*w, x^4*z^13 - y^16*w, x^5*z^7 - y^11*w, x^7 - y*z^5*w, x^6*z - y^6*w, x*y^5 - z^6]
n = 7: computation time 0.00116 seconds
[z^50 - y^49*w, x*z^43 - y^43*w, x^2*z^36 - y^37*w, x^3*z^29 - y^31*w, x^4*z^22 - y^25*w, x^5*z^15 - y^19*w, x^6*z^8 - y^13*w, x^8 - y*z^6*w, x^7*z - y^7*w, x*y^6 - z^7]
n = 8: computation time 0.00110 seconds
[z^65 - y^64*w, x*z^57 - y^57*w, x^2*z^49 - y^50*w, x^3*z^41 - y^43*w, x^4*z^33 - y^36*w, x^5*z^25 - y^29*w, x^6*z^17 - y^22*w, x^7*z^9 - y^15*w, x^9 - y*z^7*w, x^8*z - y^8*w, x*y^7 - z^8]
n = 9: computation time 0.00109 seconds
[z^82 - y^81*w, x*z^73 - y^73*w, x^2*z^64 - y^65*w, x^3*z^55 - y^57*w, x^4*z^46 - y^49*w, x^5*z^37 - y^41*w, x^6*z^28 - y^33*w, x^7*z^19 - y^25*w, x^8*z^10 - y^17*w, x^10 - y*z^8*w, x^9*z - y^9*w, x*y^8 - z^9]
n = 10: computation time 0.00108 seconds
[z^101 - y^100*w, x*z^91 - y^91*w, x^2*z^81 - y^82*w, x^3*z^71 - y^73*w, x^4*z^61 - y^64*w, x^5*z^51 - y^55*w, x^6*z^41 - y^46*w, x^7*z^31 - y^37*w, x^8*z^21 - y^28*w, x^9*z^11 - y^19*w, x^11 - y*z^9*w, x^10*z - y^10*w, x*y^9 - z^10]
n = 11: computation time 0.00153 seconds
[z^122 - y^121*w, x*z^111 - y^111*w, x^2*z^100 - y^101*w, x^3*z^89 - y^91*w, x^4*z^78 - y^81*w, x^5*z^67 - y^71*w, x^6*z^56 - y^61*w, x^7*z^45 - y^51*w, x^8*z^34 - y^41*w, x^9*z^23 - y^31*w, x^10*z^12 - y^21*w, x^12 - y*z^10*w, x^11*z - y^11*w, x*y^10 - z^11]
n = 12: computation time 0.00119 seconds
[z^145 - y^144*w, x*z^133 - y^133*w, x^2*z^121 - y^122*w, x^3*z^109 - y^111*w, x^4*z^97 - y^100*w, x^5*z^85 - y^89*w, x^6*z^73 - y^78*w, x^7*z^61 - y^67*w, x^8*z^49 - y^56*w, x^9*z^37 - y^45*w, x^10*z^25 - y^34*w, x^11*z^13 - y^23*w, x^13 - y*z^11*w, x^12*z - y^12*w, x*y^11 - z^12]
n = 13: computation time 0.00117 seconds
[z^170 - y^169*w, x*z^157 - y^157*w, x^2*z^144 - y^145*w, x^3*z^131 - y^133*w, x^4*z^118 - y^121*w, x^5*z^105 - y^109*w, x^6*z^92 - y^97*w, x^7*z^79 - y^85*w, x^8*z^66 - y^73*w, x^9*z^53 - y^61*w, x^10*z^40 - y^49*w, x^11*z^27 - y^37*w, x^12*z^14 - y^25*w, x^14 - y*z^12*w, x^13*z - y^13*w, x*y^12 - z^13]
n = 14: computation time 0.00128 seconds
[z^197 - y^196*w, x*z^183 - y^183*w, x^2*z^169 - y^170*w, x^3*z^155 - y^157*w, x^4*z^141 - y^144*w, x^5*z^127 - y^131*w, x^6*z^113 - y^118*w, x^7*z^99 - y^105*w, x^8*z^85 - y^92*w, x^9*z^71 - y^79*w, x^10*z^57 - y^66*w, x^11*z^43 - y^53*w, x^12*z^29 - y^40*w, x^13*z^15 - y^27*w, x^15 - y*z^13*w, x^14*z - y^14*w, x*y^13 - z^14]
n = 15: computation time 0.00130 seconds
[z^226 - y^225*w, x*z^211 - y^211*w, x^2*z^196 - y^197*w, x^3*z^181 - y^183*w, x^4*z^166 - y^169*w, x^5*z^151 - y^155*w, x^6*z^136 - y^141*w, x^7*z^121 - y^127*w, x^8*z^106 - y^113*w, x^9*z^91 - y^99*w, x^10*z^76 - y^85*w, x^11*z^61 - y^71*w, x^12*z^46 - y^57*w, x^13*z^31 - y^43*w, x^14*z^16 - y^29*w, x^16 - y*z^14*w, x^15*z - y^15*w, x*y^14 - z^15]
n = 16: computation time 0.00129 seconds
[z^257 - y^256*w, x*z^241 - y^241*w, x^2*z^225 - y^226*w, x^3*z^209 - y^211*w, x^4*z^193 - y^196*w, x^5*z^177 - y^181*w, x^6*z^161 - y^166*w, x^7*z^145 - y^151*w, x^8*z^129 - y^136*w, x^9*z^113 - y^121*w, x^10*z^97 - y^106*w, x^11*z^81 - y^91*w, x^12*z^65 - y^76*w, x^13*z^49 - y^61*w, x^14*z^33 - y^46*w, x^15*z^17 - y^31*w, x^17 - y*z^15*w, x^16*z - y^16*w, x*y^15 - z^16]
n = 17: computation time 0.00147 seconds
Polynomial Sequence with 20 Polynomials in 4 Variables
n = 18: computation time 0.00136 seconds
Polynomial Sequence with 21 Polynomials in 4 Variables
n = 19: computation time 0.00133 seconds
Polynomial Sequence with 22 Polynomials in 4 Variables
n = 20: computation time 0.00127 seconds
Polynomial Sequence with 23 Polynomials in 4 Variables
n = 21: computation time 0.00136 seconds
Polynomial Sequence with 24 Polynomials in 4 Variables
n = 22: computation time 0.00129 seconds
Polynomial Sequence with 25 Polynomials in 4 Variables
n = 23: computation time 0.00132 seconds
Polynomial Sequence with 26 Polynomials in 4 Variables
n = 24: computation time 0.00137 seconds
Polynomial Sequence with 27 Polynomials in 4 Variables
n = 25: computation time 0.00202 seconds
Polynomial Sequence with 28 Polynomials in 4 Variables
n = 26: computation time 0.00159 seconds
Polynomial Sequence with 29 Polynomials in 4 Variables
n = 27: computation time 0.00154 seconds
Polynomial Sequence with 30 Polynomials in 4 Variables
n = 28: computation time 0.00141 seconds
Polynomial Sequence with 31 Polynomials in 4 Variables
n = 29: computation time 0.00141 seconds
Polynomial Sequence with 32 Polynomials in 4 Variables
n = 30: computation time 0.00120 seconds
Polynomial Sequence with 33 Polynomials in 4 Variables
n = 31: computation time 0.00131 seconds
Polynomial Sequence with 34 Polynomials in 4 Variables
n = 32: computation time 0.00160 seconds
Polynomial Sequence with 35 Polynomials in 4 Variables
n = 33: computation time 0.00154 seconds
Polynomial Sequence with 36 Polynomials in 4 Variables
n = 34: computation time 0.00138 seconds
Polynomial Sequence with 37 Polynomials in 4 Variables
n = 35: computation time 0.00120 seconds
Polynomial Sequence with 38 Polynomials in 4 Variables
n = 36: computation time 0.00110 seconds
Polynomial Sequence with 39 Polynomials in 4 Variables
n = 37: computation time 0.00121 seconds
Polynomial Sequence with 40 Polynomials in 4 Variables
n = 38: computation time 0.00084 seconds
Polynomial Sequence with 41 Polynomials in 4 Variables
n = 39: computation time 0.00082 seconds
Polynomial Sequence with 42 Polynomials in 4 Variables
n = 40: computation time 0.00087 seconds
Polynomial Sequence with 43 Polynomials in 4 Variables
n = 41: computation time 0.00086 seconds
Polynomial Sequence with 44 Polynomials in 4 Variables
n = 42: computation time 0.00087 seconds
Polynomial Sequence with 45 Polynomials in 4 Variables
n = 43: computation time 0.00098 seconds
Polynomial Sequence with 46 Polynomials in 4 Variables
n = 44: computation time 0.00106 seconds
Polynomial Sequence with 47 Polynomials in 4 Variables
n = 45: computation time 0.00096 seconds
Polynomial Sequence with 48 Polynomials in 4 Variables
n = 46: computation time 0.00091 seconds
Polynomial Sequence with 49 Polynomials in 4 Variables
n = 47: computation time 0.00100 seconds
Polynomial Sequence with 50 Polynomials in 4 Variables
n = 48: computation time 0.00080 seconds
Polynomial Sequence with 51 Polynomials in 4 Variables
n = 49: computation time 0.00083 seconds
Polynomial Sequence with 52 Polynomials in 4 Variables
n = 50: computation time 0.00081 seconds
Polynomial Sequence with 53 Polynomials in 4 Variables
n = 51: computation time 0.00082 seconds
Polynomial Sequence with 54 Polynomials in 4 Variables
n = 52: computation time 0.00081 seconds
Polynomial Sequence with 55 Polynomials in 4 Variables
n = 53: computation time 0.00083 seconds
Polynomial Sequence with 56 Polynomials in 4 Variables
n = 54: computation time 0.00084 seconds
Polynomial Sequence with 57 Polynomials in 4 Variables
n = 55: computation time 0.00128 seconds
Polynomial Sequence with 58 Polynomials in 4 Variables
n = 56: computation time 0.00138 seconds
Polynomial Sequence with 59 Polynomials in 4 Variables
n = 57: computation time 0.00119 seconds
Polynomial Sequence with 60 Polynomials in 4 Variables
n = 58: computation time 0.00105 seconds
Polynomial Sequence with 61 Polynomials in 4 Variables
n = 59: computation time 0.00098 seconds
Polynomial Sequence with 62 Polynomials in 4 Variables
n = 60: computation time 0.00096 seconds
Polynomial Sequence with 63 Polynomials in 4 Variables
n = 61: computation time 0.00098 seconds
Polynomial Sequence with 64 Polynomials in 4 Variables
n = 62: computation time 0.00097 seconds
Polynomial Sequence with 65 Polynomials in 4 Variables
n = 63: computation time 0.00112 seconds
Polynomial Sequence with 66 Polynomials in 4 Variables
n = 64: computation time 0.00095 seconds
Polynomial Sequence with 67 Polynomials in 4 Variables
n = 65: computation time 0.00127 seconds
Polynomial Sequence with 68 Polynomials in 4 Variables
n = 66: computation time 0.00136 seconds
Polynomial Sequence with 69 Polynomials in 4 Variables
n = 67: computation time 0.00138 seconds
Polynomial Sequence with 70 Polynomials in 4 Variables
n = 68: computation time 0.00112 seconds
Polynomial Sequence with 71 Polynomials in 4 Variables
n = 69: computation time 0.00107 seconds
Polynomial Sequence with 72 Polynomials in 4 Variables
n = 70: computation time 0.00125 seconds
Polynomial Sequence with 73 Polynomials in 4 Variables
n = 71: computation time 0.00108 seconds
Polynomial Sequence with 74 Polynomials in 4 Variables
n = 72: computation time 0.00108 seconds
Polynomial Sequence with 75 Polynomials in 4 Variables
n = 73: computation time 0.00109 seconds
Polynomial Sequence with 76 Polynomials in 4 Variables
n = 74: computation time 0.00108 seconds
Polynomial Sequence with 77 Polynomials in 4 Variables
n = 75: computation time 0.00139 seconds
Polynomial Sequence with 78 Polynomials in 4 Variables
n = 76: computation time 0.00216 seconds
Polynomial Sequence with 79 Polynomials in 4 Variables
n = 77: computation time 0.00142 seconds
Polynomial Sequence with 80 Polynomials in 4 Variables
n = 78: computation time 0.00131 seconds
Polynomial Sequence with 81 Polynomials in 4 Variables
n = 79: computation time 0.00119 seconds
Polynomial Sequence with 82 Polynomials in 4 Variables
n = 80: computation time 0.00120 seconds
Polynomial Sequence with 83 Polynomials in 4 Variables
n = 81: computation time 0.00119 seconds
Polynomial Sequence with 84 Polynomials in 4 Variables
n = 82: computation time 0.00184 seconds
Polynomial Sequence with 85 Polynomials in 4 Variables
n = 83: computation time 0.00143 seconds
Polynomial Sequence with 86 Polynomials in 4 Variables
n = 84: computation time 0.00282 seconds
Polynomial Sequence with 87 Polynomials in 4 Variables
n = 85: computation time 0.00165 seconds
Polynomial Sequence with 88 Polynomials in 4 Variables
n = 86: computation time 0.00137 seconds
Polynomial Sequence with 89 Polynomials in 4 Variables
n = 87: computation time 0.00133 seconds
Polynomial Sequence with 90 Polynomials in 4 Variables
n = 88: computation time 0.00150 seconds
Polynomial Sequence with 91 Polynomials in 4 Variables
n = 89: computation time 0.00136 seconds
Polynomial Sequence with 92 Polynomials in 4 Variables
n = 90: computation time 0.00175 seconds
Polynomial Sequence with 93 Polynomials in 4 Variables
n = 91: computation time 0.00248 seconds
Polynomial Sequence with 94 Polynomials in 4 Variables
n = 92: computation time 0.00162 seconds
Polynomial Sequence with 95 Polynomials in 4 Variables
n = 93: computation time 0.00160 seconds
Polynomial Sequence with 96 Polynomials in 4 Variables
n = 94: computation time 0.00146 seconds
Polynomial Sequence with 97 Polynomials in 4 Variables
n = 95: computation time 0.00144 seconds
Polynomial Sequence with 98 Polynomials in 4 Variables
n = 96: computation time 0.00139 seconds
Polynomial Sequence with 99 Polynomials in 4 Variables
n = 97: computation time 0.00151 seconds
Polynomial Sequence with 100 Polynomials in 4 Variables
n = 98: computation time 0.00230 seconds
Polynomial Sequence with 101 Polynomials in 4 Variables
n = 99: computation time 0.00173 seconds
Polynomial Sequence with 102 Polynomials in 4 Variables
n = 100: computation time 0.00156 seconds
Polynomial Sequence with 103 Polynomials in 4 Variables
n = 101: computation time 0.00155 seconds
Polynomial Sequence with 104 Polynomials in 4 Variables
n = 102: computation time 0.00150 seconds
Polynomial Sequence with 105 Polynomials in 4 Variables
n = 103: computation time 0.00166 seconds
Polynomial Sequence with 106 Polynomials in 4 Variables
n = 104: computation time 0.00205 seconds
Polynomial Sequence with 107 Polynomials in 4 Variables
n = 105: computation time 0.00228 seconds
Polynomial Sequence with 108 Polynomials in 4 Variables
n = 106: computation time 0.00189 seconds
Polynomial Sequence with 109 Polynomials in 4 Variables
n = 107: computation time 0.00170 seconds
Polynomial Sequence with 110 Polynomials in 4 Variables
n = 108: computation time 0.00184 seconds
Polynomial Sequence with 111 Polynomials in 4 Variables
n = 109: computation time 0.00171 seconds
Polynomial Sequence with 112 Polynomials in 4 Variables
n = 110: computation time 0.00174 seconds
Polynomial Sequence with 113 Polynomials in 4 Variables
n = 111: computation time 0.00284 seconds
Polynomial Sequence with 114 Polynomials in 4 Variables
n = 112: computation time 0.00200 seconds
Polynomial Sequence with 115 Polynomials in 4 Variables
n = 113: computation time 0.00194 seconds
Polynomial Sequence with 116 Polynomials in 4 Variables
n = 114: computation time 0.00182 seconds
Polynomial Sequence with 117 Polynomials in 4 Variables
n = 115: computation time 0.00177 seconds
Polynomial Sequence with 118 Polynomials in 4 Variables
n = 116: computation time 0.00211 seconds
Polynomial Sequence with 119 Polynomials in 4 Variables
n = 117: computation time 0.00233 seconds
Polynomial Sequence with 120 Polynomials in 4 Variables
n = 118: computation time 0.00200 seconds
Polynomial Sequence with 121 Polynomials in 4 Variables
n = 119: computation time 0.00187 seconds
Polynomial Sequence with 122 Polynomials in 4 Variables
n = 120: computation time 0.00184 seconds
Polynomial Sequence with 123 Polynomials in 4 Variables
n = 121: computation time 0.00208 seconds
Polynomial Sequence with 124 Polynomials in 4 Variables
n = 122: computation time 0.00236 seconds
Polynomial Sequence with 125 Polynomials in 4 Variables
n = 123: computation time 0.00233 seconds
Polynomial Sequence with 126 Polynomials in 4 Variables
n = 124: computation time 0.00202 seconds
Polynomial Sequence with 127 Polynomials in 4 Variables
n = 125: computation time 0.00213 seconds
Polynomial Sequence with 128 Polynomials in 4 Variables
n = 126: computation time 0.00196 seconds
Polynomial Sequence with 129 Polynomials in 4 Variables
n = 127: computation time 0.00214 seconds
Polynomial Sequence with 130 Polynomials in 4 Variables
n = 128: computation time 0.00222 seconds
Polynomial Sequence with 131 Polynomials in 4 Variables
n = 129: computation time 0.00226 seconds
Polynomial Sequence with 132 Polynomials in 4 Variables
n = 130: computation time 0.00218 seconds
Polynomial Sequence with 133 Polynomials in 4 Variables
n = 131: computation time 0.00208 seconds
Polynomial Sequence with 134 Polynomials in 4 Variables
n = 132: computation time 0.00211 seconds
Polynomial Sequence with 135 Polynomials in 4 Variables
n = 133: computation time 0.00246 seconds
Polynomial Sequence with 136 Polynomials in 4 Variables
n = 134: computation time 0.00257 seconds
Polynomial Sequence with 137 Polynomials in 4 Variables
n = 135: computation time 0.00224 seconds
Polynomial Sequence with 138 Polynomials in 4 Variables
n = 136: computation time 0.00222 seconds
Polynomial Sequence with 139 Polynomials in 4 Variables
n = 137: computation time 0.00282 seconds
Polynomial Sequence with 140 Polynomials in 4 Variables
n = 138: computation time 0.00290 seconds
Polynomial Sequence with 141 Polynomials in 4 Variables
n = 139: computation time 0.00259 seconds
Polynomial Sequence with 142 Polynomials in 4 Variables
n = 140: computation time 0.00245 seconds
Polynomial Sequence with 143 Polynomials in 4 Variables
n = 141: computation time 0.00250 seconds
Polynomial Sequence with 144 Polynomials in 4 Variables
n = 142: computation time 0.00257 seconds
Polynomial Sequence with 145 Polynomials in 4 Variables
n = 143: computation time 0.00262 seconds
Polynomial Sequence with 146 Polynomials in 4 Variables
n = 144: computation time 0.00249 seconds
Polynomial Sequence with 147 Polynomials in 4 Variables
n = 145: computation time 0.00260 seconds
Polynomial Sequence with 148 Polynomials in 4 Variables
n = 146: computation time 0.00241 seconds
Polynomial Sequence with 149 Polynomials in 4 Variables
n = 147: computation time 0.00277 seconds
Polynomial Sequence with 150 Polynomials in 4 Variables
n = 148: computation time 0.00272 seconds
Polynomial Sequence with 151 Polynomials in 4 Variables
n = 149: computation time 0.00254 seconds
Polynomial Sequence with 152 Polynomials in 4 Variables
n = 150: computation time 0.00252 seconds
Polynomial Sequence with 153 Polynomials in 4 Variables
n = 151: computation time 0.00316 seconds
Polynomial Sequence with 154 Polynomials in 4 Variables
n = 152: computation time 0.00284 seconds
Polynomial Sequence with 155 Polynomials in 4 Variables
n = 153: computation time 0.00268 seconds
Polynomial Sequence with 156 Polynomials in 4 Variables
n = 154: computation time 0.00275 seconds
Polynomial Sequence with 157 Polynomials in 4 Variables
n = 155: computation time 0.00319 seconds
Polynomial Sequence with 158 Polynomials in 4 Variables
n = 156: computation time 0.00309 seconds
Polynomial Sequence with 159 Polynomials in 4 Variables
n = 157: computation time 0.00291 seconds
Polynomial Sequence with 160 Polynomials in 4 Variables
n = 158: computation time 0.00273 seconds
Polynomial Sequence with 161 Polynomials in 4 Variables
n = 159: computation time 0.00330 seconds
Polynomial Sequence with 162 Polynomials in 4 Variables
n = 160: computation time 0.00314 seconds
Polynomial Sequence with 163 Polynomials in 4 Variables
n = 161: computation time 0.00283 seconds
Polynomial Sequence with 164 Polynomials in 4 Variables
n = 162: computation time 0.00279 seconds
Polynomial Sequence with 165 Polynomials in 4 Variables
n = 163: computation time 0.00339 seconds
Polynomial Sequence with 166 Polynomials in 4 Variables
n = 164: computation time 0.00312 seconds
Polynomial Sequence with 167 Polynomials in 4 Variables
n = 165: computation time 0.00299 seconds
Polynomial Sequence with 168 Polynomials in 4 Variables
n = 166: computation time 0.00313 seconds
Polynomial Sequence with 169 Polynomials in 4 Variables
n = 167: computation time 0.00325 seconds
Polynomial Sequence with 170 Polynomials in 4 Variables
n = 168: computation time 0.00325 seconds
Polynomial Sequence with 171 Polynomials in 4 Variables
n = 169: computation time 0.00329 seconds
Polynomial Sequence with 172 Polynomials in 4 Variables
n = 170: computation time 0.00311 seconds
Polynomial Sequence with 173 Polynomials in 4 Variables
n = 171: computation time 0.00330 seconds
Polynomial Sequence with 174 Polynomials in 4 Variables
n = 172: computation time 0.00341 seconds
Polynomial Sequence with 175 Polynomials in 4 Variables
n = 173: computation time 0.00319 seconds
Polynomial Sequence with 176 Polynomials in 4 Variables
n = 174: computation time 0.00320 seconds
Polynomial Sequence with 177 Polynomials in 4 Variables
n = 175: computation time 0.00408 seconds
Polynomial Sequence with 178 Polynomials in 4 Variables
n = 176: computation time 0.00334 seconds
Polynomial Sequence with 179 Polynomials in 4 Variables
n = 177: computation time 0.00354 seconds
Polynomial Sequence with 180 Polynomials in 4 Variables
n = 178: computation time 0.00381 seconds
Polynomial Sequence with 181 Polynomials in 4 Variables
n = 179: computation time 0.00358 seconds
Polynomial Sequence with 182 Polynomials in 4 Variables
n = 180: computation time 0.00382 seconds
Polynomial Sequence with 183 Polynomials in 4 Variables
n = 181: computation time 0.00361 seconds
Polynomial Sequence with 184 Polynomials in 4 Variables
n = 182: computation time 0.00364 seconds
Polynomial Sequence with 185 Polynomials in 4 Variables
n = 183: computation time 0.00349 seconds
Polynomial Sequence with 186 Polynomials in 4 Variables
n = 184: computation time 0.00363 seconds
Polynomial Sequence with 187 Polynomials in 4 Variables
n = 185: computation time 0.00383 seconds
Polynomial Sequence with 188 Polynomials in 4 Variables
n = 186: computation time 0.00364 seconds
Polynomial Sequence with 189 Polynomials in 4 Variables
n = 187: computation time 0.00363 seconds
Polynomial Sequence with 190 Polynomials in 4 Variables
n = 188: computation time 0.00359 seconds
Polynomial Sequence with 191 Polynomials in 4 Variables
n = 189: computation time 0.00375 seconds
Polynomial Sequence with 192 Polynomials in 4 Variables
n = 190: computation time 0.00372 seconds
Polynomial Sequence with 193 Polynomials in 4 Variables
n = 191: computation time 0.00368 seconds
Polynomial Sequence with 194 Polynomials in 4 Variables
n = 192: computation time 0.00400 seconds
Polynomial Sequence with 195 Polynomials in 4 Variables
n = 193: computation time 0.00412 seconds
Polynomial Sequence with 196 Polynomials in 4 Variables
n = 194: computation time 0.00379 seconds
Polynomial Sequence with 197 Polynomials in 4 Variables
n = 195: computation time 0.00400 seconds
Polynomial Sequence with 198 Polynomials in 4 Variables
n = 196: computation time 0.00398 seconds
Polynomial Sequence with 199 Polynomials in 4 Variables
n = 197: computation time 0.00442 seconds
Polynomial Sequence with 200 Polynomials in 4 Variables
n = 198: computation time 0.00406 seconds
Polynomial Sequence with 201 Polynomials in 4 Variables
n = 199: computation time 0.00399 seconds
Polynomial Sequence with 202 Polynomials in 4 Variables
n = 200: computation time 0.00476 seconds
Polynomial Sequence with 203 Polynomials in 4 Variables
In [39]:
for g in G:
    print(g)

z^40001 - y^40000*w
x*z^39801 - y^39801*w
x^2*z^39601 - y^39602*w
x^3*z^39401 - y^39403*w
x^4*z^39201 - y^39204*w
x^5*z^39001 - y^39005*w
x^6*z^38801 - y^38806*w
x^7*z^38601 - y^38607*w
x^8*z^38401 - y^38408*w
x^9*z^38201 - y^38209*w
x^10*z^38001 - y^38010*w
x^11*z^37801 - y^37811*w
x^12*z^37601 - y^37612*w
x^13*z^37401 - y^37413*w
x^14*z^37201 - y^37214*w
x^15*z^37001 - y^37015*w
x^16*z^36801 - y^36816*w
x^17*z^36601 - y^36617*w
x^18*z^36401 - y^36418*w
x^19*z^36201 - y^36219*w
x^20*z^36001 - y^36020*w
x^21*z^35801 - y^35821*w
x^22*z^35601 - y^35622*w
x^23*z^35401 - y^35423*w
x^24*z^35201 - y^35224*w
x^25*z^35001 - y^35025*w
x^26*z^34801 - y^34826*w
x^27*z^34601 - y^34627*w
x^28*z^34401 - y^34428*w
x^29*z^34201 - y^34229*w
x^30*z^34001 - y^34030*w
x^31*z^33801 - y^33831*w
x^32*z^33601 - y^33632*w
x^33*z^33401 - y^33433*w
x^34*z^33201 - y^33234*w
x^35*z^33001 - y^33035*w
x^36*z^32801 - y^32836*w
x^37*z^32601 - y^32637*w
x^38*z^32401 - y^32438*w
x^39*z^32201 - y^32239*w
x^40*z^32001 - y^32040*w
x^41*z^31801 - y^31841*w
x^42*z^31601 - y^31642*w
x^43*z^31401 - y^31443*w
x^44*z^31201 - y^31244*w
x^45*z^31001 - y^31045*w
x^46*z^30801 - y^30846*w
x^47*z^30601 - y^30647*w
x^48*z^30401 - y^30448*w
x^49*z^30201 - y^30249*w
x^50*z^30001 - y^30050*w
x^51*z^29801 - y^29851*w
x^52*z^29601 - y^29652*w
x^53*z^29401 - y^29453*w
x^54*z^29201 - y^29254*w
x^55*z^29001 - y^29055*w
x^56*z^28801 - y^28856*w
x^57*z^28601 - y^28657*w
x^58*z^28401 - y^28458*w
x^59*z^28201 - y^28259*w
x^60*z^28001 - y^28060*w
x^61*z^27801 - y^27861*w
x^62*z^27601 - y^27662*w
x^63*z^27401 - y^27463*w
x^64*z^27201 - y^27264*w
x^65*z^27001 - y^27065*w
x^66*z^26801 - y^26866*w
x^67*z^26601 - y^26667*w
x^68*z^26401 - y^26468*w
x^69*z^26201 - y^26269*w
x^70*z^26001 - y^26070*w
x^71*z^25801 - y^25871*w
x^72*z^25601 - y^25672*w
x^73*z^25401 - y^25473*w
x^74*z^25201 - y^25274*w
x^75*z^25001 - y^25075*w
x^76*z^24801 - y^24876*w
x^77*z^24601 - y^24677*w
x^78*z^24401 - y^24478*w
x^79*z^24201 - y^24279*w
x^80*z^24001 - y^24080*w
x^81*z^23801 - y^23881*w
x^82*z^23601 - y^23682*w
x^83*z^23401 - y^23483*w
x^84*z^23201 - y^23284*w
x^85*z^23001 - y^23085*w
x^86*z^22801 - y^22886*w
x^87*z^22601 - y^22687*w
x^88*z^22401 - y^22488*w
x^89*z^22201 - y^22289*w
x^90*z^22001 - y^22090*w
x^91*z^21801 - y^21891*w
x^92*z^21601 - y^21692*w
x^93*z^21401 - y^21493*w
x^94*z^21201 - y^21294*w
x^95*z^21001 - y^21095*w
x^96*z^20801 - y^20896*w
x^97*z^20601 - y^20697*w
x^98*z^20401 - y^20498*w
x^99*z^20201 - y^20299*w
x^100*z^20001 - y^20100*w
x^101*z^19801 - y^19901*w
x^102*z^19601 - y^19702*w
x^103*z^19401 - y^19503*w
x^104*z^19201 - y^19304*w
x^105*z^19001 - y^19105*w
x^106*z^18801 - y^18906*w
x^107*z^18601 - y^18707*w
x^108*z^18401 - y^18508*w
x^109*z^18201 - y^18309*w
x^110*z^18001 - y^18110*w
x^111*z^17801 - y^17911*w
x^112*z^17601 - y^17712*w
x^113*z^17401 - y^17513*w
x^114*z^17201 - y^17314*w
x^115*z^17001 - y^17115*w
x^116*z^16801 - y^16916*w
x^117*z^16601 - y^16717*w
x^118*z^16401 - y^16518*w
x^119*z^16201 - y^16319*w
x^120*z^16001 - y^16120*w
x^121*z^15801 - y^15921*w
x^122*z^15601 - y^15722*w
x^123*z^15401 - y^15523*w
x^124*z^15201 - y^15324*w
x^125*z^15001 - y^15125*w
x^126*z^14801 - y^14926*w
x^127*z^14601 - y^14727*w
x^128*z^14401 - y^14528*w
x^129*z^14201 - y^14329*w
x^130*z^14001 - y^14130*w
x^131*z^13801 - y^13931*w
x^132*z^13601 - y^13732*w
x^133*z^13401 - y^13533*w
x^134*z^13201 - y^13334*w
x^135*z^13001 - y^13135*w
x^136*z^12801 - y^12936*w
x^137*z^12601 - y^12737*w
x^138*z^12401 - y^12538*w
x^139*z^12201 - y^12339*w
x^140*z^12001 - y^12140*w
x^141*z^11801 - y^11941*w
x^142*z^11601 - y^11742*w
x^143*z^11401 - y^11543*w
x^144*z^11201 - y^11344*w
x^145*z^11001 - y^11145*w
x^146*z^10801 - y^10946*w
x^147*z^10601 - y^10747*w
x^148*z^10401 - y^10548*w
x^149*z^10201 - y^10349*w
x^150*z^10001 - y^10150*w
x^151*z^9801 - y^9951*w
x^152*z^9601 - y^9752*w
x^153*z^9401 - y^9553*w
x^154*z^9201 - y^9354*w
x^155*z^9001 - y^9155*w
x^156*z^8801 - y^8956*w
x^157*z^8601 - y^8757*w
x^158*z^8401 - y^8558*w
x^159*z^8201 - y^8359*w
x^160*z^8001 - y^8160*w
x^161*z^7801 - y^7961*w
x^162*z^7601 - y^7762*w
x^163*z^7401 - y^7563*w
x^164*z^7201 - y^7364*w
x^165*z^7001 - y^7165*w
x^166*z^6801 - y^6966*w
x^167*z^6601 - y^6767*w
x^168*z^6401 - y^6568*w
x^169*z^6201 - y^6369*w
x^170*z^6001 - y^6170*w
x^171*z^5801 - y^5971*w
x^172*z^5601 - y^5772*w
x^173*z^5401 - y^5573*w
x^174*z^5201 - y^5374*w
x^175*z^5001 - y^5175*w
x^176*z^4801 - y^4976*w
x^177*z^4601 - y^4777*w
x^178*z^4401 - y^4578*w
x^179*z^4201 - y^4379*w
x^180*z^4001 - y^4180*w
x^181*z^3801 - y^3981*w
x^182*z^3601 - y^3782*w
x^183*z^3401 - y^3583*w
x^184*z^3201 - y^3384*w
x^185*z^3001 - y^3185*w
x^186*z^2801 - y^2986*w
x^187*z^2601 - y^2787*w
x^188*z^2401 - y^2588*w
x^189*z^2201 - y^2389*w
x^190*z^2001 - y^2190*w
x^191*z^1801 - y^1991*w
x^192*z^1601 - y^1792*w
x^193*z^1401 - y^1593*w
x^194*z^1201 - y^1394*w
x^195*z^1001 - y^1195*w
x^196*z^801 - y^996*w
x^197*z^601 - y^797*w
x^198*z^401 - y^598*w
x^199*z^201 - y^399*w
x^201 - y*z^199*w
x^200*z - y^200*w
x*y^199 - z^200
In [ ]: