Computing the maximum likelihood degree of a four-way scaled Segre variety

July 29, 2020
Isaac Burke

Using Macaulay2, we may compute the maximum likelihood degree (“ML degree”) of a scaled Segre variety where $d=4$ and $n_1=n_2=n_3=n_4$. We will express the scaling array \begin{equation} C=[c_{ijkl}] \in (\mathbb{C}^*)^{[2] \times [2] \times [2] \times [2]} \end{equation} as a row vector with the entries ordered lexicographically.

The Macaulay2 commands for twenty-four distinct scaling arrays $C$ are given below (hover cursor over a code box and click to copy). The commands may be executed with a local version of Macaulay2 or by using one of the online Macaulay2 applications here.

ML degree 1: when $C=(3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
3*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
3*s*t_11*t_21*t_41,
1*s*t_11*t_21,
3*s*t_11*t_31*t_41,
1*s*t_11*t_31,
3*s*t_11*t_41,
1*s*t_11,
3*s*t_21*t_31*t_41,
1*s*t_21*t_31,
3*s*t_21*t_41,
1*s*t_21,
3*s*t_31*t_41,
1*s*t_31,
3*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 2: when $C=(1,1,1,1,1,1,1,1,2,2,2,2,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
2*s*t_21*t_31*t_41,
2*s*t_21*t_31,
2*s*t_21*t_41,
2*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 3: when $C=(3,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
3*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
3*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 4: when $C=(2,4,2,2,1,2,1,1,1,2,1,1,1,2,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
4*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
2*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
2*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 5: when $C=(3,1,1,2,1,1,1,1,3,1,1,2,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
3*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
2*s*t_11*t_21,
1*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
3*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
2*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 6: when $C=(1,1,2,1,1,2,2,1,1,1,2,1,1,1,2,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
2*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 7: when $C=(1,2,1,1,1,2,1,1,1,2,1,1,2,3,2,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
2*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
2*s*t_31*t_41,
3*s*t_31,
2*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 8: when $C=(1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 9: when $C=(1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
2*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
2*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 10: when $C=(1,2,1,1,1,2,2,1,1,2,1,1,1,1,2,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
2*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
2*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 11: when $C=(2,1,1,2,2,1,1,1,1,2,1,1,1,2,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
2*s*t_11*t_21,
2*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
2*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 12: when $C=(2,1,1,1,1,2,2,1,1,1,2,1,1,2,2,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
2*s*t_31,
2*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 13: when $C=(2,1,1,2,2,1,1,1,1,2,2,1,1,2,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
2*s*t_11*t_21,
2*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
2*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 14: when $C=(1,2,2,1,2,1,1,1,1,1,2,2,1,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
2*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
1*s*t_11*t_21,
2*s*t_11*t_31*t_41,
1*s*t_11*t_31,
1*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
2*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 15: when $C=(2,1,2,1,3,1,2,1,2,1,3,1,2,1,3,3)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
1*s*t_11*t_21,
3*s*t_11*t_31*t_41,
1*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
2*s*t_21*t_31*t_41,
1*s*t_21*t_31,
3*s*t_21*t_41,
1*s*t_21,
2*s*t_31*t_41,
1*s*t_31,
3*s*t_41,
3*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 16: when $C=(1,2,2,1,1,2,2,1,2,1,1,1,1,1,1,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
2*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
2*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 17: when $C=(2,1,1,1,1,2,2,1,1,2,1,1,1,1,2,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
2*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 18: when $C=(2,1,1,1,1,2,2,1,1,1,1,1,2,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
2*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 19: when $C=(2,1,1,1,1,2,2,1,1,1,1,1,1,1,1,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 20: when $C=(1,2,2,1,1,1,2,1,2,1,1,1,1,1,1,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
1*s*t_11*t_21*t_31*t_41,
2*s*t_11*t_21*t_31,
2*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
1*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
2*s*t_21*t_31*t_41,
1*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 21: when $C=(2,1,1,1,1,2,2,1,1,1,2,1,2,1,1,1)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
2*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 22: when $C=(2,1,1,1,1,2,2,1,1,1,2,1,1,1,1,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
1*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 23: when $C=(2,1,3,2,1,2,2,1,3,2,1,1,2,1,2,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
3*s*t_11*t_21*t_41,
2*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
3*s*t_21*t_31*t_41,
2*s*t_21*t_31,
1*s*t_21*t_41,
1*s*t_21,
2*s*t_31*t_41,
1*s*t_31,
2*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I

ML degree 24: when $C=(2,1,1,1,1,2,2,1,1,2,2,1,1,1,1,2)$:

R = QQ[s,t_11,t_21,t_31,t_41];
sT = transpose matrix{{
2*s*t_11*t_21*t_31*t_41,
1*s*t_11*t_21*t_31,
1*s*t_11*t_21*t_41,
1*s*t_11*t_21,
1*s*t_11*t_31*t_41,
2*s*t_11*t_31,
2*s*t_11*t_41,
1*s*t_11,
1*s*t_21*t_31*t_41,
2*s*t_21*t_31,
2*s*t_21*t_41,
1*s*t_21,
1*s*t_31*t_41,
1*s*t_31,
1*s*t_41,
2*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
{1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0},
{1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0},
{1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0},
{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}};
U = transpose matrix{{
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999),
random(0,999),random(0,999)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I