# Great system of quadratic equations.

I have a great system of quadratic equations to solve. The variables are $$Co[i,j,k]$$ with 3 indices. Here for convenience I denote $$x_l = Co[i,j,k]$$, where l is decimal and (i, j, k) is ternary.

``````                Do[{i, j, k} = {0, 0, IntegerDigits[l, 3][[1]]};
Co[i, j, k] = Subindex[x, l], {l, 0, 2, 1}]Do[{i, j, k} = {0, IntegerDigits[l, 3][[1]],
Whole digits[l, 3][[2]]};
Co[i, j, k] = Subindex[x, l]{1, 3, 8, 1}]Do[{ijk}={IntegerDigits[{ijk}={IntegerDigits[{ijk}={IntegerDigits[{ijk}={IntegerDigits[l, 3][[1]], IntegerDigits[l, 3][[2]],
Whole digits[l, 3][[3]]};
Co[i, j, k] = Subindex[x, l], {l, 9, 26, 1}]
``````

here $$eq_ {lam}$$ They are the equations to solve.

``````                Do[lam = j 3^3 + k 3^2 + n 3 + s;
Subscript[eq, lam] =
Sum[Co[j, k, m]*Co[m, n, s] - Co[k, n, m]*Co[j, m, s], {m, 0, 2, 1}],
{j, 0, 2, 1}, {k, 0, 2, 1}, {n, 0, 2, 1}, {s, 0, 2, 1}]
``````

One can, for example, print all the equations.

``````                Do[Print[Subscript["Equation", i]]; Print[Subscript[eq, i]], {i, 0,80,1}]
``````

I have 81 equations and 27 variables. $$x_0, x_1, ldots, x_ {26}$$. Yes $$vec {x}$$ Then it is a vector of solutions. $$const * , vec {x}$$ It is also a solution (even for $$const = 0$$), therefore, one can establish a restriction $$x ^ 2_0 + x ^ 2_1 + ldots + x ^ 2_ {26} = 1$$

Here I try to solve the system (without restriction):

``````                Solve[
Subscript[eq, 1] == 0 && Subindex[eq, 2] == 0 &&
Subscript[eq, 3] == 0 && Subindex[eq, 4] == 0 &&
Subscript[eq, 5] == 0 && Subindex[eq, 6] == 0 &&
Subscript[eq, 7] == 0 && Subindex[eq, 8] == 0 &&
Subscript[eq, 9] == 0 && Subindex[eq, 10] == 0 &&
Subscript[eq, 11] == 0 && Subindex[eq, 12] == 0 &&
Subscript[eq, 13] == 0 && Subindex[eq, 14] == 0 &&
Subscript[eq, 15] == 0 && Subindex[eq, 16] == 0 &&
Subscript[eq, 17] == 0 && Subindex[eq, 18] == 0 &&
Subscript[eq, 19] == 0 && Subindex[eq, 20] == 0 &&
Subscript[eq, 21] == 0 && Subindex[eq, 22] == 0 &&
Subscript[eq, 23] == 0 && Subindex[eq, 24] == 0 &&
Subscript[eq, 25] == 0 && Subindex[eq, 26] == 0 &&
Subscript[eq, 27] == 0 && Subindex[eq, 28] == 0 &&
Subscript[eq, 29] == 0 && Subindex[eq, 30] == 0 &&
Subscript[eq, 31] == 0 && Subindex[eq, 32] == 0 &&
Subscript[eq, 33] == 0 && Subindex[eq, 34] == 0 &&
Subscript[eq, 35] == 0 && Subindex[eq, 36] == 0 &&
Subscript[eq, 37] == 0 && Subindex[eq, 38] == 0 &&
Subscript[eq, 39] == 0 && Subindex[eq, 40] == 0 &&
Subscript[eq, 41] == 0 && Subindex[eq, 42] == 0 &&
Subscript[eq, 43] == 0 && Subindex[eq, 44] == 0 &&
Subscript[eq, 45] == 0 && Subindex[eq, 46] == 0 &&
Subscript[eq, 47] == 0 && Subindex[eq, 48] == 0 &&
Subscript[eq, 49] == 0 && Subindex[eq, 50] == 0 &&
Subscript[eq, 51] == 0 && Subindex[eq, 52] == 0 &&
Subscript[eq, 53] == 0 && Subindex[eq, 54] == 0 &&
Subscript[eq, 55] == 0 && Subindex[eq, 56] == 0 &&
Subscript[eq, 57] == 0 && Subindex[eq, 58] == 0 &&
Subscript[eq, 59] == 0 && Subindex[eq, 60] == 0 &&
Subscript[eq, 61] == 0 && Subindex[eq, 62] == 0 &&
Subscript[eq, 63] == 0 && Subindex[eq, 64] == 0 &&
Subscript[eq, 65] == 0 && Subindex[eq, 66] == 0 &&
Subscript[eq, 67] == 0 && Subindex[eq, 68] == 0 &&
Subscript[eq, 69] == 0 && Subindex[eq, 70] == 0 &&
Subscript[eq, 71] == 0 && Subindex[eq, 72] == 0 &&
Subscript[eq, 73] == 0 && Subindex[eq, 74] == 0 &&
Subscript[eq, 75] == 0 && Subindex[eq, 76] == 0 &&
Subscript[eq, 77] == 0 && Subindex[eq, 78] == 0 &&
Subscript[eq, 79] == 0 && Subindex[eq, 80] == 0 &&
Subscript[eq, 0] == 0
, {Subindex[x, 0], Subscript[x, 1], Subscript[x, 2], Subscript[x,
3], Subscript[x, 4], Subscript[x, 5], Subscript[x, 6], Subscript[x,
7], Subscript[x, 8], Subscript[x, 9], Subscript[x, 10], Subscript[x,
11], Subscript[x, 12], Subscript[x, 13], Subscript[x, 14],
Subscript[x, 15], Subscript[x, 16], Subscript[x, 17], Subscript[x,
18], Subscript[x, 19], Subscript[x, 20], Subscript[x, 21],
Subscript[x, 22], Subscript[x, 23], Subscript[x, 24], Subscript[x,
25], Subscript[x, 26]}]
``````

The calculation time is too long. Is it possible to solve the system in some way?