1935.sagews

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)))
variables

[[z41, z42, z43, z44], [z31, z32, z33, z34], [z21, z22, z23, z24], [z11, z12, z13, z14]]

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);M

[ 1 0 0 0] [z31 1 0 0] [z21 z22 1 0] [z11 z12 z13 1]

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;
e

[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]

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

[0 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 0]

a=[];
for row in range(n):
    a.append([])
    for column in range(row+3):
        a[row].append(var('a'+str(row+1)+str(column+1)))
a

[[a11, a12, a13], [a21, a22, a23, a24], [a31, a32, a33, a34, a35], [a41, a42, a43, a44, a45, a46]]

X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+3): #changed for h(i)=i+2 it was j in range (i+2): for peterson
        X[i]=X[i]+a[i][j]*M*e[j]
X

[(a11, a11*z31 + a12, a11*z21 + a12*z22 + a13, a11*z11 + a12*z12 + a13*z13), (a21, a21*z31 + a22, a21*z21 + a22*z22 + a23, a21*z11 + a22*z12 + a23*z13 + a24)]

for j in range(n-2):
    for i in range(n):
        print(e[i]*N*M*e[j],e[i]*X[j])

(z31, a11) (z21, a11*z31 + a12) (z11, a11*z21 + a12*z22 + a13) (0, a11*z11 + a12*z12 + a13*z13) (1, a21) (z22, a21*z31 + a22) (z12, a21*z21 + a22*z22 + a23) (0, a21*z11 + a22*z12 + a23*z13 + a24)

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

a11 = z31;
a12 = z21 -a11*z31;
a13 = z11 - a11*z21-a12*z22;
a11*z11 + a12*z12+a13*z13

-(z31^2 - z21)*z12 + ((z31^2 - z21)*z22 - z21*z31 + z11)*z13 + z11*z31

f1 = singular('-(z31^2 - z21)*z12 + ((z31^2 - z21)*z22 - z21*z31 + z11)*z13 + z11*z31')

I = singular.ideal(f1);I

z31*z11+z21*z12+z11*z13-z31^2*z12-z31*z21*z13-z21*z22*z13+z31^2*z22*z13

J = singular.std(I);J

z31*z11+z21*z12+z11*z13-z31^2*z12-z31*z21*z13-z21*z22*z13+z31^2*z22*z13

singular.mult(J);

2