subalgebraBasis -- Compute subalgebra basis generators

• Usage:

  N = subalgebraBasis M

  N = subalgebraBasis A

  N = subalgebraBasis L

  N = subalgebraBasis B

• Inputs:
  • A, a subring,
  • M, a matrix, of generators for a subring of a polynomial ring
  • L, a list, containing generators for a subring of a polynomial ring
  • B, a SAGBIBasis, containing a partial computation of a subalgebra basis
• Optional inputs:
  • AutoSubduce => a Boolean value, default value true, a flag indicating when to perform autosubduction on the generators before performing the subalgebra basis computation
  • ReduceNewGenerators => a Boolean value, default value true, a flag indicating whether to reduce (via Gaussian elimination) to new subalgebra basis generators before adding them to the current subalgebra basis
  • StorePending => a Boolean value, default value true, a flag that indicates whether the pending list should be stored in the result.
  • Strategy => a string, default value "Master", the name of the update strategy at the beginning of each loop: "DegreeByDegree", "Incremental", and "Master". The strategy "Master" is a hybrid method that combines the other two; starting with "DegreeByDegree" for low degrees and switching to "Incremental".
  • SubductionMethod => a string, default value "Top", the name of the method used for subduction either: "Top" or "Engine".
  • Limit => an integer, default value 20, a degree limit for the binomial S-pairs that are computed internally.
  • AutoSubduceOnPartialCompletion => a Boolean value, default value false, a flag that indicates whether autosubduction is applied to the subalgebra basis generators the first time no new generators are added. Use this only if very few new subalgebra basis generators are expected.
  • PrintLevel => an integer, default value 0, an option to produce additional output. When this is greater than zero, information is printed about the progress of the computation
  • Recompute => a Boolean value, default value false, a flag that indicates if the computation will resume on subsequent runs, otherwise it starts at the beginning
  • RenewOptions => a Boolean value, default value false, a flag that indicates if the computation will use the options specified, otherwise it will use the previously selected options (except for the following, which may always be specified: PrintLevel, Limit, Recompute, and RenewOptions)
• Outputs:
  • N, a matrix, whose entries form a partial subalgebra basis

i1 : R = QQ[t_(1,1)..t_(3,3),MonomialOrder=>Lex];

i2 : M = genericMatrix(R,3,3);

             3      3
o2 : Matrix R  <-- R

i3 : A = subring gens minors(2, M);

i4 : subalgebraBasis A

o4 = | t_(2,2)t_(3,3)-t_(2,3)t_(3,2) t_(2,1)t_(3,3)-t_(2,3)t_(3,1)
     ------------------------------------------------------------------------
     t_(2,1)t_(3,2)-t_(2,2)t_(3,1) t_(1,2)t_(3,3)-t_(1,3)t_(3,2)
     ------------------------------------------------------------------------
     t_(1,2)t_(2,3)-t_(1,3)t_(2,2) t_(1,1)t_(3,3)-t_(1,3)t_(3,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(3,2)-t_(1,2)t_(3,1) t_(1,1)t_(2,3)-t_(1,3)t_(2,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(2,2)-t_(1,2)t_(2,1)
     ------------------------------------------------------------------------
     t_(1,1)t_(2,2)t_(3,1)t_(3,3)-t_(1,1)t_(2,3)t_(3,1)t_(3,2)-t_(1,2)t_(2,1)
     ------------------------------------------------------------------------
     t_(3,1)t_(3,3)+t_(1,2)t_(2,3)t_(3,1)^2+t_(1,3)t_(2,1)t_(3,1)t_(3,2)-t_(1
     ------------------------------------------------------------------------
     ,3)t_(2,2)t_(3,1)^2 t_(1,1)t_(1,3)t_(2,2)t_(3,3)-t_(1,1)t_(1,3)t_(2,3)t
     ------------------------------------------------------------------------
     _(3,2)-t_(1,2)t_(1,3)t_(2,1)t_(3,3)+t_(1,2)t_(1,3)t_(2,3)t_(3,1)+t_(1,3
     ------------------------------------------------------------------------
     )^2t_(2,1)t_(3,2)-t_(1,3)^2t_(2,2)t_(3,1) |

             1      11
o4 : Matrix R  <-- R