How is the formulation in src/solver.cpp (lines 76–102) derived?

[Immortalqx]

Hi, thank you for your excellent work and for open-sourcing this project.

While studying the implementation, I came across the following code block in src/solver.cpp (lines 76–102):

    const std::vector<int> coeff_ind0 = {0, 6,  12, 1,  7,  13, 2,  8, 0,  6,  12, 14, 6, 0, 12, 1,  7,  13, 3,
                                         9, 2,  8,  14, 15, 4,  10, 7, 1,  16, 13, 8,  2, 6, 12, 0,  14, 9,  3,
                                         8, 14, 2,  15, 3,  9,  15, 4, 10, 16, 7,  13, 1, 5, 11, 10, 4,  17, 16};
    const std::vector<int> coeff_ind1 = {11, 17, 5, 9, 15, 3, 5, 11, 17, 10, 16, 4, 11, 5, 17};
    const std::vector<int> ind0 = {0,   1,   9,   12,  13,  21,  24,  25,  26,  28,  29,  33,  39,  42,  47,
                                   50,  52,  53,  60,  61,  62,  64,  65,  69,  72,  73,  75,  78,  81,  83,
                                   87,  90,  91,  92,  94,  95,  99,  102, 103, 104, 106, 107, 110, 112, 113,
                                   122, 124, 125, 127, 128, 130, 132, 133, 135, 138, 141, 143};
    const std::vector<int> ind1 = {7, 8, 10, 19, 20, 22, 26, 28, 29, 31, 32, 34, 39, 42, 47};
    Eigen::MatrixXd C0 = Eigen::MatrixXd::Zero(12, 12);
    Eigen::MatrixXd C1 = Eigen::MatrixXd::Zero(12, 4);

    for (int k = 0; k < ind0.size(); k++) {
        int i = ind0[k] % 12;
        int j = ind0[k] / 12;
        C0(i, j) = coeffs[coeff_ind0[k]];
    }

    for (int k = 0; k < ind1.size(); k++) {
        int i = ind1[k] % 12;
        int j = ind1[k] / 12;
        C1(i, j) = coeffs[coeff_ind1[k]];
    }

    Eigen::MatrixXd C2 = C0.colPivHouseholderQr().solve(C1);
    Eigen::Matrix4d AM;
    AM << Eigen::RowVector4d(0, 0, 1, 0), -C2.row(9), -C2.row(10), -C2.row(11);

I’ve read the associated paper and the references, but I’m still having trouble understanding how the construction of the matrices C0, C1is derived — especially how the index vectors (coeff_ind0, ind0, etc.) correspond to the mathematical formulation.

Could you please clarify:

• What mathematical derivation leads to the formulation of C0 and C1?
• How the index mappings (like ind0, coeff_ind0) were determined?
• Currently, my understanding is that C0 and C1 correspond to the matrices $C_R$ and $C_B$ from the paper "Efficient Solvers for Minimal Problems by Syzygy-based Reduction". If that is the case, why is the matrix $C_E$ not needed or omitted in the implementation?

Any pointers to a specific part of the paper or an external reference would also be greatly appreciated.

Thanks again for your time and help!