Solve telling me all my solutions are zero

Question

I'm trying to use Mathematica to solve a complicated system of equations. Here is a screenshot of the equations, and what Mathematica outputs.

The variables $v_i$ and $w_i$ act as constants in this equation. I do understand the solution Mathematica gives is technically correct, however I do not want the trivial solution. How can I make Mathematica ignore that particular solution?

This is the code, as asked for.

ybe[a_, b_, c_, d_, e_, f_, R_, S_, T_] :=
  Sum[Subscript[R, d, e, \[Alpha], \[Beta]] Subscript[S, \[Alpha], f, a, \[Gamma]] Subscript[T, \[Beta], \[Gamma], b, c] - Subscript[T, e, f, \[Beta], \[Gamma]] Subscript[S, d, \[Gamma], \[Alpha], c] Subscript[R, \[Alpha], \[Beta], a, b], {\[Alpha], 0, 1}, {\[Beta], 0,1}, {\[Gamma], 0, 1}]

Subscript[r[u_], k_, l_, i_, j_] := 
  \[Delta][i+j, k+l] (\[Delta][i, j, k, l, 0] Subscript[u, 1] + \[Delta][i, j, k, l, 1] Subscript[u, 2] + \[Delta][i, k, 0] \[Delta][j, l, 1] Subscript[u, 3] + \[Delta][i, k, 1] \[Delta][j, l, 0] Subscript[u, 4] + \[Delta][i, l, 1] \[Delta][j, k, 0] Subscript[u, 5] + \[Delta][i, l ,0] \[Delta][j, k, 1] Subscript[u, 6])

eqns = DeleteCases[
   Flatten[
     Table[ybe[a, b, c, d, e, f, r[u], r[v], r[w]] == 0,
        {a, 0, 1}, {b, 0, 1}, {c, 0, 1}, {d, 0, 1}, {e, 0, 1}, {f, 0, 1}]],
   0 == 0]

Solve[eqns, Table[Subscript[u, i], {i, 6}]]

Answer 1

There are two issues. One is that you did not provide the correct code for generating the equations. Waste of time for anyone who, like myself, tried to work on this (an immediate conclusion being, this wasted MY time).

Here is correct code.

ybe[a_, b_, c_, d_, e_, f_, R_, S_, T_] := 
 Sum[Subscript[R, d, e, \[Alpha], \[Beta]] Subscript[S, \[Alpha], f, 
     a, \[Gamma]] Subscript[T, \[Beta], \[Gamma], b, c] - 
   Subscript[T, e, f, \[Beta], \[Gamma]] Subscript[S, 
     d, \[Gamma], \[Alpha], c] Subscript[R, \[Alpha], \[Beta], a, 
     b], {\[Alpha], 0, 1}, {\[Beta], 0, 1}, {\[Gamma], 0, 1}]

Subscript[r[u_], k_, l_, i_, 
  j_] := \[Delta][i + j, 
    k + l] (\[Delta][i, j, k, l, 0] Subscript[u, 1] + \[Delta][i, j, 
       k, l, 1] Subscript[u, 2] + \[Delta][i, k, 0] \[Delta][j, l, 
       1] Subscript[u, 3] + \[Delta][i, k, 1] \[Delta][j, l, 
       0] Subscript[u, 4] + \[Delta][i, l, 1] \[Delta][j, k, 
       0] Subscript[u, 5] + \[Delta][i, l, 0] \[Delta][j, k, 
       1] Subscript[u, 6]) /. \[Delta] -> KroneckerDelta

exprs = DeleteCases[
   Flatten[Table[
     ybe[a, b, c, d, e, f, r[u], r[v], r[w]], {a, 0, 1}, {b, 0, 
      1}, {c, 0, 1}, {d, 0, 1}, {e, 0, 1}, {f, 0, 1}]], 0];

Exercise: What was the necessary change that corrects this?

The second issue is less obvious. All solutions other than the origin are nongeneric in that they force equations on some "constants". To get an idea of what these are one can enlarge the variable list e.g. by throwing the v variables into it.

vars = Join[Table[Subscript[v, i], {i, 6}], 
   Table[Subscript[u, i], {i, 6}]];

solns = Solve[exprs == 0, vars];

This will give some idea of what happens in the solution set.

Length[solns]

(* Out[199]= 36 *)

solns[[1 ;; 4]]

(* Out[203]= {{Subscript[u, 1] -> 0, Subscript[u, 2] -> 0, 
  Subscript[u, 3] -> 0, Subscript[u, 4] -> 0, Subscript[u, 5] -> 0, 
  Subscript[u, 6] -> 0}, {Subscript[u, 5] -> 0, Subscript[u, 6] -> 0, 
  Subscript[v, 3] -> (Subscript[u, 3] Subscript[v, 1])/Subscript[u, 
   1], Subscript[v, 4] -> (Subscript[u, 4] Subscript[v, 2])/Subscript[
   u, 2], Subscript[v, 5] -> 0, 
  Subscript[v, 6] -> 0}, {Subscript[v, 1] -> 0, Subscript[v, 2] -> 0, 
  Subscript[v, 3] -> 0, Subscript[v, 4] -> 0, Subscript[v, 5] -> 0, 
  Subscript[v, 6] -> 0}, {Subscript[u, 1] -> 0, Subscript[u, 3] -> 0, 
  Subscript[u, 5] -> 0, Subscript[u, 6] -> 0, 
  Subscript[v, 4] -> (Subscript[u, 4] Subscript[v, 2])/Subscript[u, 
   2], Subscript[v, 5] -> 0, Subscript[v, 6] -> 0}} *)

Some are uglier than those. Here is one such.

(* {Subscript[u, 1] -> (1/(
  2 Subscript[v, 5] Subscript[w, 
   1]))(Subscript[u, 4] Subscript[v, 5] Subscript[w, 3] + (
    Subscript[u, 4] Subscript[v, 5] Subscript[w, 1] Subscript[w, 2])/
    Subscript[w, 4] - (
    Subscript[u, 4] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])/
    Subscript[w, 4] - 1/(Subscript[u, 5] Subscript[w, 4])(\[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
          Subscript[w, 4] Subscript[w, 5] Subscript[w, 
         6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
            Subscript[w, 1] Subscript[w, 2] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            3] Subscript[w, 4] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            5] Subscript[w, 6])^2))), 
 Subscript[u, 2] -> (1/(
  2 Subscript[v, 5] Subscript[w, 
   2]))((Subscript[u, 3] Subscript[v, 5] Subscript[w, 1] Subscript[w, 
     2])/Subscript[w, 3] + 
    Subscript[u, 3] Subscript[v, 5] Subscript[w, 4] - (
    Subscript[u, 3] Subscript[v, 5] Subscript[w, 5] Subscript[w, 6])/
    Subscript[w, 3] - 
    1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 3])Subscript[u, 
     3] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
          Subscript[w, 4] Subscript[w, 5] Subscript[w, 
         6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
            Subscript[w, 1] Subscript[w, 2] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            3] Subscript[w, 4] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            5] Subscript[w, 6])^2)), Subscript[u, 6] -> 0, 
 Subscript[v, 1] -> (1/(2 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) Subscript[w, 4]
    Subscript[w, 
   5]))(Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 
     1] Subscript[w, 2] - 
    Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 3]
      Subscript[w, 4] - 
    Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 5]
      Subscript[w, 6] - \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
         Subscript[w, 4] Subscript[w, 5] Subscript[w, 
        6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
           Subscript[w, 1] Subscript[w, 2] + 
         Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 
          3] Subscript[w, 4] + 
         Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w, 
          5] Subscript[w, 6])^2)), 
 Subscript[v, 2] -> (1/(
  2 Subscript[u, 4] Subscript[w, 
   3]))((Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 
     1] Subscript[w, 2])/(Subscript[u, 5] Subscript[w, 5]) - (
    Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 3]
      Subscript[w, 4])/(Subscript[u, 5] Subscript[w, 5]) - (
    Subscript[u, 3] Subscript[u, 4] Subscript[v, 5] Subscript[w, 6])/
    Subscript[u, 5] - 1/(\!\(
\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ 
\*SubscriptBox[\(w\), \(5\)]\))Subscript[u, 3] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
          Subscript[w, 4] Subscript[w, 5] Subscript[w, 
         6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
            Subscript[w, 1] Subscript[w, 2] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            3] Subscript[w, 4] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            5] Subscript[w, 6])^2)), 
 Subscript[v, 3] -> (1/(
  2 Subscript[u, 5] Subscript[w, 
   2]))((Subscript[u, 3] Subscript[v, 5] Subscript[w, 1] Subscript[w, 
     2])/Subscript[w, 5] - (
    Subscript[u, 3] Subscript[v, 5] Subscript[w, 3] Subscript[w, 4])/
    Subscript[w, 5] + 
    Subscript[u, 3] Subscript[v, 5] Subscript[w, 6] - 
    1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 5])Subscript[u, 
     3] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
          Subscript[w, 4] Subscript[w, 5] Subscript[w, 
         6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
            Subscript[w, 1] Subscript[w, 2] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            3] Subscript[w, 4] + 
          Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[w,
            5] Subscript[w, 6])^2)), 
 Subscript[v, 4] -> ((2 Subscript[u, 4] 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(w\), \(1\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(w\), \(2\), \(2\)]\))/(Subscript[u, 5] 
\!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) - (4 Subscript[u, 4] 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 1]
       Subscript[w, 2] Subscript[w, 3] Subscript[w, 4])/(
     Subscript[u, 5] 
\!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) + (2 Subscript[u, 4] 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(w\), \(3\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(w\), \(4\), \(2\)]\))/(Subscript[u, 5] 
\!\(\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\)) - (2 Subscript[u, 4] 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 1]
       Subscript[w, 2] Subscript[w, 6])/(
     Subscript[u, 5] Subscript[w, 5]) - (2 Subscript[u, 4] 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
       Subscript[w, 4] Subscript[w, 6])/(
     Subscript[u, 5] Subscript[w, 5]) - 1/(\!\(
\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ 
\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\))2 Subscript[v, 5]
       Subscript[w, 1] Subscript[w, 2] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
           Subscript[w, 4] Subscript[w, 5] Subscript[w, 
          6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
             Subscript[w, 1] Subscript[w, 2] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 3] Subscript[w, 4] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 5] Subscript[w, 6])^2) + 1/(\!\(
\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\ 
\*SubsuperscriptBox[\(w\), \(5\), \(2\)]\))2 Subscript[v, 5]
       Subscript[w, 3] Subscript[w, 4] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
           Subscript[w, 4] Subscript[w, 5] Subscript[w, 
          6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
             Subscript[w, 1] Subscript[w, 2] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 3] Subscript[w, 4] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 5] Subscript[w, 6])^2))/((2 Subscript[v, 5] 
\!\(\*SubsuperscriptBox[\(w\), \(1\), \(2\)]\) Subscript[w, 2])/
     Subscript[w, 5] - (
     2 Subscript[v, 5] Subscript[w, 1] Subscript[w, 3] Subscript[w, 
      4])/Subscript[w, 5] - 
     2 Subscript[v, 5] Subscript[w, 1] Subscript[w, 6] - 
     1/(Subscript[u, 4] Subscript[u, 5] Subscript[w, 5])2 Subscript[w,
       1] \[Sqrt](-4 
\!\(\*SubsuperscriptBox[\(u\), \(4\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(u\), \(5\), \(2\)]\) 
\!\(\*SubsuperscriptBox[\(v\), \(5\), \(2\)]\) Subscript[w, 3]
           Subscript[w, 4] Subscript[w, 5] Subscript[w, 
          6] + (-Subscript[u, 4] Subscript[u, 5] Subscript[v, 5]
             Subscript[w, 1] Subscript[w, 2] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 3] Subscript[w, 4] + 
           Subscript[u, 4] Subscript[u, 5] Subscript[v, 5] Subscript[
            w, 5] Subscript[w, 6])^2)), Subscript[v, 6] -> 0} *)