CoCalc -- 2009.sagews

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n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[1,2,3,4]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):



        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');
print a;
b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z21,z22,z23,z31,z32,z41,z42)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

[
[a11 == z31, a12 == -z31^2 + z21, a21 == 1, a22 == z22 - z31, a23 == -z22^2 + z22*z31 + z12 - z21]
]
 Equations defining neighborhood:
-z11+z21*z22+z21*z31-z22*z31^2,
z12*z21+z11*z31-z12*z31^2,
z11+z12*z13-z13*z21+z12*z22-z12*z31-z13*z22^2+z13*z22*z31
Multiplicity at fixed point:
4
Reduced:
True
Primary Decomposition:
[Ideal (z22*z31^2 - z21*z22 - z21*z31 + z11, z12*z31^2 - z12*z21 - z11*z31, z12*z21*z31 - z11*z22*z31 - z11*z12, z13*z22^2 - z13*z22*z31 - z12*z13 + z13*z21 - z12*z22 + z12*z31 - z11, z12*z21^2 - z11*z21*z22 - z11*z12*z31 + z11^2, z13*z21*z22*z31 - z13*z21^2 - z11*z13*z22 - z11*z22*z31 + z11*z21, z13*z21^2*z22 - z11*z13*z22*z31 - z11*z13*z21 - z11*z21*z22 + z11^2, z13*z21^3 - 2*z11*z13*z21*z31 + z11^2*z13 - z11*z21^2 + z11^2*z31) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[1, 2, 3, 4]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[2,1,3,4]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
-z11+z22+z21^2-z21*z22*z41,
z12+z11*z21-z12*z21*z41,
z12*z13+z11*z22-z13*z21*z22-z12*z22*z41+z13*z22^2*z41
Multiplicity at fixed point:
2
Reduced:
True
Primary Decomposition:
[Ideal (z21*z22*z41 - z21^2 + z11 - z22, z12*z21*z41 - z11*z21 - z12, z11*z12*z41 - z11^2 - z12*z21 + z11*z22, z12*z21^2 - z11*z21*z22 - z11*z12, z12*z13*z21 - z11*z13*z22 + z13*z22^2 - z12*z22, z13*z22^2*z41 - z13*z21*z22 - z12*z22*z41 + z12*z13 + z11*z22, z11*z13*z22*z41 - z11*z13*z21 - z13*z21*z22 + z11*z22, z13*z21*z22^2 + z11*z12*z13 - z12*z21*z22, z13*z21^2*z22 + z11^2*z13 - z11*z13*z22 - z11*z21*z22, z13*z21^3 + z11^2*z13*z41 - 2*z11*z13*z21 - z11*z21^2 + z11^2, z11*z13*z22^3 - z13*z22^4 + z11*z12^2*z13 - z12^2*z21*z22 + z12*z22^3) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[2, 1, 3, 4]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[1,3,2,4]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
-z21+z31^2+z11*z32-z21*z31*z32,
z11*z12+z11*z31-z12*z21*z31,
z13+z12^2+z11*z32-z12*z13*z32-z12*z21*z32-z13*z31*z32+z13*z21*z32^2
Multiplicity at fixed point:
2
Reduced:
True
Primary Decomposition:
[Ideal (z21*z31*z32 - z31^2 - z11*z32 + z21, z12*z31^2 - z11*z31*z32 - z12*z21, z12*z21*z31 - z11*z12 - z11*z31, z12*z21^2 - z11*z12*z31 + z11^2*z32 - z11*z21, z13*z21*z32^2 - z12*z13*z32 - z12*z21*z32 - z13*z31*z32 + z12^2 + z11*z32 + z13, z12*z13*z31*z32 - z11*z13*z32^2 - z12^2*z31 + z11*z12*z32 + z13*z21*z32 - z13*z31, z13*z21^2*z32 - z11*z12*z31 - z13*z21*z31 + z11^2*z32 + z11*z13, z12*z13*z21*z32 - z12^2*z21 + z11*z13*z32 - z13*z21, z11*z13*z21*z32 - z11*z12*z21 - z13*z21^2 + z11*z13*z31, z12^2*z13*z32 + 2*z11*z13*z32^2 - z12^3 + z12^2*z31 - 2*z11*z12*z32 - 2*z13*z21*z32 - z12*z13 + z13*z31, z11*z12*z13*z32 + 2*z11*z13*z31*z32 - z11*z12^2 - z11*z12*z31 - z13*z21*z31, z13*z21^2*z31 - 2*z11*z13*z31^2 - z11^2*z13*z32 + z11^2*z12 + z11*z13*z21 + z11^2*z31, z13*z21^3 - 2*z11*z13*z21*z31 + z11^2*z13 + z11^2*z21, 2*z11*z13*z31^2*z32 + z11^2*z13*z32^2 - z13*z21*z31^2 - z11^2*z12*z32 - z11^2*z31*z32 - 2*z11*z12*z21 - z13*z21^2 + 2*z11*z13*z31) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[1, 3, 2, 4]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[1,2,4,3]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
z12*z21+z11*z31-z12*z31^2,
-z11+z21*z22+z21*z31-z22*z31^2,
-z12+z21+z22^2-z11*z23-z22*z31-z12*z22*z23+z12*z23*z31
Multiplicity at fixed point:
2
Reduced:
True
Primary Decomposition:
[Ideal (z22*z31^2 - z21*z22 - z21*z31 + z11, z12*z31^2 - z12*z21 - z11*z31, z12*z21*z31 - z11*z22*z31 - z11*z12, z12*z22*z23 - z12*z23*z31 - z22^2 + z11*z23 + z22*z31 + z12 - z21, z12*z21^2 - z11*z21*z22 - z11*z12*z31 + z11^2, z11*z22*z23*z31 - z11*z21*z23 - z21*z22*z31 + z21^2 + z11*z22, z11*z21*z22*z23 - z21^2*z22 - z11^2*z23 + z11*z22*z31 + z11*z21, z11*z21^2*z23 - z11^2*z23*z31 - z21^3 + 2*z11*z21*z31 - z11^2) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[1, 2, 4, 3]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[2,1,4,3]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
z12+z11*z21-z12*z21*z41,
-z11+z22+z21^2-z21*z22*z41,
-z12+z21*z22-z11*z22*z23-z22^2*z41+z12*z22*z23*z41
Multiplicity at fixed point:
2
Reduced:
True
Primary Decomposition:
[Ideal (z21*z22*z41 - z21^2 + z11 - z22, z12*z21*z41 - z11*z21 - z12, z11*z12*z41 - z11^2 - z12*z21 + z11*z22, z12*z22*z23 - z12*z21 + z11*z22 - z22^2, z11*z22*z23 + z11*z22*z41 - z11*z21 - z21*z22, z12*z21^2 - z11*z21*z22 - z11*z12, z11*z21^2*z23 - z21^3 - z11^2*z23 - z11^2*z41 + 2*z11*z21) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[2, 1, 4, 3]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[3,2,1,4]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
-z31+z42+z11*z41-z11*z31*z42,
z12+z11^2-z11*z12*z31,
z13+z11*z12-z12^2*z31-z12*z13*z41+z12*z13*z31*z42
Multiplicity at fixed point:
1
Reduced:
True
Primary Decomposition:
[Ideal (2*z13*z31*z42 - z13*z42^2 - z12*z31 - z13*z41 + z12*z42, z11*z31*z42 - z11*z41 + z31 - z42, z11*z13*z42 - z11*z12 - z13, z11*z13*z41 - z11^2 - 2*z13*z31 + z13*z42 - z12, z11*z12*z41 - z11^2*z42 - z12*z31, z12*z31^2 - z11*z31 - z12*z41 + z11*z42, z12*z13*z31 - z12*z13*z42 + z12^2 - z11*z13, z11*z12*z31 - z11^2 - z12, z12*z13*z42^2 - z12^2*z31 - z12*z13*z41 - z12^2*z42 + 2*z11*z12 + 2*z13) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[3, 2, 1, 4]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[1,4,3,2]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
-z11+z21*z31-z11*z22*z31,
-z21+z31^2-z11*z31*z32,
-z22+z33+z31*z32-z11*z32^2-z21*z32*z33+z11*z22*z32*z33
Multiplicity at fixed point:
1
Reduced:
True
Primary Decomposition:
[Ideal (z21*z22 - z11*z32 - z21*z33, z11*z32*z33 + z22*z31 - z21*z32 - z31*z33, z11*z21*z33 - z21^2 + z11*z31, z11*z31*z32 - z31^2 + z21, z22*z31^2 - z21*z31*z32 - z21*z33, z22^2*z31 - z22*z31*z33 - z31*z32 + z22 - z33, z11*z22*z31 - z21*z31 + z11, z21^2*z31*z32 - z21*z31^2*z33 - z31^3 + z21^2*z33 + z21*z31) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[1, 4, 3, 2]

n=4;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))


p=[4,3,2,1]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;


s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)


a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))


X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]


poly=[];
for j in range(n-2):
    poly.append([]);
    for i in range(n):
        poly[j].append(e[i]*N*M*e[j] == e[i]*X[j]);

polyls=[];
polyls2=[];
solvestring=''
for row in range (n-2):
    for i in range(row+2):
        polyls.append(poly[row][p[i]-1]);
        if not(row==0) or not (i==0): 
            solvestring=solvestring+','
        solvestring=solvestring+('a'+str(row+1)+str(i+1))
    for j in range(row+2,n):


        polyls2.append(poly[row][p[j]-1]);
        #solvestring=solvestring+(',z'+str(n-p[j])+str(row+1));
exec('a=solve(polyls,'+solvestring+')');

b1=polyls2[0].subs(a[0][0])
b2=b1.subs(a[0][1]);
#print b2;
b2l=-b2.lhs();
b2r=b2.rhs();
#print b2l; print b2r;
b3=polyls2[1].subs(a[0][0])
b4=b3.subs(a[0][1]);
#print b4;
b4l=-b4.lhs(); b4r = b4.rhs();

R=singular.ring(0,'(z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43)','ds');

f1 = singular(b2l+b2r);
#print f1;
I = singular.ideal(str(f1));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;

f2=singular(b4l+b4r);
#print f2;
I = singular.ideal(str(f1),str(f2));I
J = singular.std(I);J
mult =singular.mult(J);
#print  mult;


c1=polyls2[2].subs(a[0][2])
c2=c1.subs(a[0][3]);
c3=c2.subs(a[0][4]);
#print c3;
c3l=-c3.lhs(); c3r=c3.rhs();
#print(c3l); print(c3r);

f3=singular(c3l+c3r);
I = singular.ideal(str(f1),str(f2),str(f3)); 
print " Equations defining neighborhood:";
print I;
J = singular.std(I);J
mult =singular.mult(J);
print "Multiplicity at fixed point:"
print mult;

R.<z11,z12,z13,z14,z15,z21,z22,z23,z24,z31,z32,z33,z41,z42,z43> = QQ[];

A=R.ideal(f1,f2,f3);
AR=A.radical();
Reduced = AR == A; 

print "Reduced:";
print Reduced;

AP = A.primary_decomposition();
print "Primary Decomposition:";
print AP;

print "Permutation:";
print p;

Equations defining neighborhood:
-z21+z32,
-z31+z42,
-z32+z43
Multiplicity at fixed point:
1
Reduced:
True
Primary Decomposition:
[Ideal (z32 - z43, z31 - z42, z21 - z43) of Multivariate Polynomial Ring in z11, z12, z13, z14, z15, z21, z22, z23, z24, z31, z32, z33, z41, z42, z43 over Rational Field]
Permutation:
[4, 3, 2, 1]