Given a list of 24 polynomials over $\mathbb{Z}[x,y,z]$ namely Lf, computed by sage using the code
R.<x,y,z>=QQ['x, y, z']
I=(x*y^3-z)*R
Q= R.quo(I)
N=15
p=3
q=5
L=[(0, 0, 0),(0, 0, 1),(0, 0, 2),(0, 0, 3),(0, 0, 4),(0, 0, 5),(0, 0, 6),(0, 0, 7),(0, 1, 0),(0, 1, 1),(0, 1, 2),(0, 2, 0),(0,2,1),
(0, 2, 2),(1, 0, 0),(1, 0, 1),(1, 0, 2),(1, 0, 3),(1, 0, 4),(1, 1, 0),(1, 1, 1),(1, 1, 2),(2, 0, 0),(2, 0, 1)]
X=floor(N^2)
Y=floor(N^2)
Z=X*Y^3
phi=((p^4-1)*(q^4-1))/((p-1)*(q-1))
e=79
d=79
f=x*(y^3+(N+1)*y^2+(N^2-2*N+1)*y+N^3-N^2-N+1)+1
M=Matrix(0,len(L))
for (k,i,j) in L:
g=Q(x^i*y^j*f^k*e^(2-k)).lift()
h=g(x*X,y*Y,z*Z)
L1=vector(h[x^i*y^j*z^k] for (k,i,j) in L)
M = M.stack(L1)
N=M.LLL()
L2=[x^i*y^j*z^k for (k,i,j) in L]
r=0
Lf=[]
Li=[]
for i in srange(24):
D=N[i,:]*transpose(matrix([L2]))
r=D[0,0]
Li=[r]
Lf.extend(Li)
print(Lf)
In the theory, I know that there are three polynomials from the list Lf say $p$, $q$, and $r$ having the same root (x,y,z), to extract such root, it suffices to compute the grobner basis of the three polynomials and solving the the system of polynomials $p(x,y,z)=q(x,y,z)=r(x,y,z)=0$, so I have to find the three polynomials from Grobner basis. I have computed these by sage code, the grobener basis here is denoted E
for i in srange(24):
for j in srange(i+1,24):
for k in srange(j+1,24):
J = R.ideal([Lf[i],Lf[j],Lf[k]])
E=J.groebner_basis()
if len(E)==3:
print(E)
So far, my problem is how to sovle these polynomials ?
