Using Macaulay2, we may compute the maximum likelihood degree (“ML degree”) of a scaled Segre variety where $d=2$ and $n_1=n_2=3$ and the scaling array is given by

\begin{equation} C = \left[\begin{array}{ccc} c_{13} & c_{23} & c_{33} \\ c_{12} & c_{22} & c_{32} \\ c_{11} & c_{21} & c_{31} \end{array} \right] = \left[\begin{array}{ccc} c_{13} & 1 & 1 \\ c_{12} & c_{22} & 1 \\ 1 & c_{21} & 1 \end{array} \right]. \end{equation}

The Macaulay2 commands for six 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 $(c_{12},c_{13},c_{21},c_{22})=(1,1,1,1)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
1*s*t_11*t_22,
1*s*t_11,
1*s*t_12*t_21,
1*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
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 $(c_{12},c_{13},c_{21},c_{22})=(2,1,1,1)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
2*s*t_11*t_22,
1*s*t_11,
1*s*t_12*t_21,
1*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
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 $(c_{12},c_{13},c_{21},c_{22})=(2,3,1,1)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
2*s*t_11*t_22,
3*s*t_11,
1*s*t_12*t_21,
1*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
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 $(c_{12},c_{13},c_{21},c_{22})=(2,3,1,2)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
2*s*t_11*t_22,
3*s*t_11,
1*s*t_12*t_21,
2*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
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 $(c_{12},c_{13},c_{21},c_{22})=(2,3,2,1)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
2*s*t_11*t_22,
3*s*t_11,
2*s*t_12*t_21,
1*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
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 $(c_{12},c_{13},c_{21},c_{22})=(2,3,2,3)$:

R = QQ[s,t_11,t_12,t_21,t_22];
sT = transpose matrix{{
1*s*t_11*t_21,
2*s*t_11*t_22,
3*s*t_11,
2*s*t_12*t_21,
3*s*t_12*t_22,
1*s*t_12,
1*s*t_21,
1*s*t_22,
1*s}};
As = matrix{
{1,1,1,1,1,1,1,1,1},
{1,1,1,0,0,0,0,0,0},
{0,0,0,1,1,1,0,0,0},
{1,0,0,1,0,0,1,0,0},
{0,1,0,0,1,0,0,1,0}};
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)}};
As = substitute(As,R);
U = substitute(U,R);
Usum = (sum(flatten(entries U)));
rels = As*U-Usum*As*sT;
I = ideal rels
degree I