subduction -- Subduction against a set elements

Synopsis

• Usage:

  N = subduction(SB, M)

  n = subduction(SB, m)

  N = subduction(S, M)

  n = subduction(S, m)

  N = subduction(F, M)

  n = subduction(F, m)

  nList = subduction(FList, mList)

  n = subduction(FList, m)

• Inputs:
  • SB, a SAGBIBasis, whose subalgebra generators are subducted against
  • S, a subring, whose generators are subducted against
  • F, a matrix, whose entries are subducted against
  • FList, a list, whose entries are subducted against
  • M, a matrix, a 1-row matrix whose entries are to be subducted
  • m, a ring element, the element to be subducted
  • mList, a list, a list of elements to be subducted
• 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 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 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 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, apply autosubduction to the subalgebra generators the first time no new generators are added. Use this only if very few new subalgebra generators are expected.
  • PrintLevel => an integer, default value 0, When this is greater than zero, information is printed about the progress of the computation
  • Recompute => a Boolean value, default value false, if true then the computation will resume, otherwise it starts at the beginning
  • RenewOptions => a Boolean value, default value false, if true then the computation uses the options specified, otherwise it uses the previously selected options (except for the following, which may always be specified: PrintLevel, Limit, Recompute, and RenewOptions)
• Outputs:
  • N, a matrix, entries of $M$ after subduction
  • n, a ring element, the element $m$ after subduction
  • nList, a list, the elements of $mList$ after subduction

Description

i1 : R = QQ[x,y];

i2 : S = subring {x+y, x*y, x*y^2};

i3 : f = x^4;

i4 : subduction(S, f)

           3    4
o4 = - 2x*y  - y

o4 : R

i5 : sagbi S;

i6 : subduction(gens sagbi S, f)

       4
o6 = -y

o6 : R

See also

• sagbi -- Compute a subalgebra basis (sagbi basis)
• Subring -- The type of all subrings
• SAGBIBasis -- The type of all subalgebra bases
• RingElement % Subring -- Remainder modulo a subring
• AutoSubduce -- Flag for autosubduction before the sagbi algorithm
• ReduceNewGenerators -- Flag for reducing new generators in Sagbi algorithm
• StorePending -- Flag for storing the pending list to the result of the Sagbi algorithm
• SubductionMethod -- Subduction method for the Sagbi algorithm
• AutoSubduceOnPartialCompletion -- Subduct subalgebra generators at the end of the sagbi algorithm
• PrintLevel -- Levels of information displayed during Sagbi algorithm
• Recompute -- Flag for restarting a sagbi or isSAGBI computation
• RenewOptions -- Flag for reselecting the options for a subalgebra bases computation
• Subduction computation limit -- Bound for subalgebra basis computation
• Subduction strategies -- Update procedure for the Sagbi algorithm
• groebnerSubductionQuotient -- Extrinsic method for subduction quotients
• RingElement // Subring -- Subduction quotient with respect to a subring