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1.2.20. Generic Minors
The following example computes the relations among the 2x2 minors of
a generic 2xN matrix for a range of values of N.  Note the use of
indeterminates with multiple indices.

Example

Use R ::= Q[t,x,y,z];
Define Det_SubAlgebra(N)
  L := [];
  For C1 := 1 To N-1 Do
    For C2 := C1+1 To N Do
      P := y[C1,C2]-(x[1,C1] x[2,C2] - x[2,C1] x[1,C2]);
      Append(L,P)
    EndFor
  Endfor;
  Return Ideal(L)
EndDefine;
Define Det_SubAlgebra_Print(N)  -- calculate and print relations
  J := Det_SubAlgebra(N);
  PrintLn NewLine,'N = ',N;
  PrintLn 'Sub-algebra equations:';
  PrintLn Gens(Elim(x,J))
EndDefine;
Set Indentation;
For N := 3 To 5 Do
  S ::= Z/(32003)[y[1..(N-1),2..N],x[1..2,1..N]];
  Using S Do
    Det_SubAlgebra_Print(N);
  EndUsing
EndFor;

N = 3
Sub-algebra equations:
[
  0]

N = 4
Sub-algebra equations:
[
  2y[1,4]y[2,3] - 2y[1,3]y[2,4] + 2y[1,2]y[3,4]]

N = 5
Sub-algebra equations:
[
  2y[2,5]y[3,4] - 2y[2,4]y[3,5] + 2y[2,3]y[4,5],
  2y[1,5]y[3,4] - 2y[1,4]y[3,5] + 2y[1,3]y[4,5],
  2y[1,5]y[2,4] - 2y[1,4]y[2,5] + 2y[1,2]y[4,5],
  2y[1,5]y[2,3] - 2y[1,3]y[2,5] + 2y[1,2]y[3,5],
  2y[1,4]y[2,3] - 2y[1,3]y[2,4] + 2y[1,2]y[3,4]]

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