1902.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=[1,2,4,3]
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);M

[ 1 0 0 0] [z31 1 0 0] [z21 z22 z23 1] [z11 z12 1 0]

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+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))
a

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

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]
X

[(a11, a11*z31 + a12, a11*z21 + a12*z22, a11*z11 + a12*z12), (a21, a21*z31 + a22, a21*z21 + a22*z22 + a23*z23, a21*z11 + a22*z12 + a23)]

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

print(poly)

[z31 == a11, z21 == a11*z31 + a12, z11 == a11*z21 + a12*z22, 0 == a11*z11 + a12*z12, 1 == a21, z22 == a21*z31 + a22, z12 == a21*z21 + a22*z22 + a23*z23, 0 == a21*z11 + a22*z12 + a23]

print(solve(poly[0],a11)); 
print(solve(poly[1],a12));
print(poly[2]);
print(poly[3]);

[ a11 == z31 ] [ a12 == -a11*z31 + z21 ] z11 == a11*z21 + a12*z22 0 == a11*z11 + a12*z12

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

0 == -(z31^2 - z21)*z12 - ((z31^2 - z21)*z22 - z21*z31)*z31

print(solve(poly[4],a21));
print(solve(poly[5],a22));
print(solve(poly[7],a23));
poly[6]

[ a21 == 1 ] [ a22 == -a21*z31 + z22 ] [ a23 == -a21*z11 - a22*z12 ] z12 == a21*z21 + a22*z22 + a23*z23

a21 = 1;
a22 = -a21*z31 + z22;
a23 = -a21*z11 - a22*z12;
z12 == a21*z21 + a22*z22 + a23*z23

z12 == (z22 - z31)*z22 - ((z22 - z31)*z12 - (z31^2 - z21)*z22 + z21*z31)*z23 + z21

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

f1 = singular('-z12 + (z22 - z31)*z22 - ((z22 - z31)*z12 - (z31^2 - z21)*z22 +z21*z31)*z23 + z21');
f2 = singular('-(z31^2 - z21)*z12 - ((z31^2 - z21)*z22 - z21*z31)*z31');
f2

z12*z21+z21*z22*z31-z12*z31^2+z21*z31^2-z22*z31^3

I = singular.ideal(f1,f2); I

-z12+z21+z22^2-z22*z31-z12*z22*z23-z21*z22*z23+z12*z23*z31-z21*z23*z31+z22*z23*z31^2, z12*z21+z21*z22*z31-z12*z31^2+z21*z31^2-z22*z31^3

J = singular.std(I);

J

z12-z21-z22^2+z22*z31+z12*z22*z23+z21*z22*z23-z12*z23*z31+z21*z23*z31-z22*z23*z31^2, z21^2+z21*z22^2-z12*z31^2+z21*z31^2-z12*z21*z22*z23-z21^2*z22*z23+z12*z21*z23*z31-z21^2*z23*z31-z22*z31^3+z21*z22*z23*z31^2

singular.mult(J);

2

print("This says that the multiplicity of e_{1243} is 2")

This says that the multiplicity of e_{1243} is 2