subduction -- Subduction of polynomials

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, an instance of the type SAGBIBasis, subduct with respect to the sagbi generators of $SB$
  • S, an instance of the type Subring, subduct with respect to the generators of $S$
  • F, a matrix, subduct with respect to the entries of $F$
  • FList, a list, subduct with respect to the entries of $FList$
  • 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 Sagbi basis computation (See: AutoSubduce)
  • ReduceNewGenerators => a Boolean value, default value true, a flag indicating whether to apply gaussian elimination to new sagbi generators before adding them to the current sagbi basis (See: ReduceNewGenerators)
  • StorePending => a Boolean value, default value true, a flag that indicates whether the pending list should be stored in the result. (See: StorePending)
  • 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 that combines the other two; starting with \"DegreeByDegree\" for low degrees and switching to \"Incremental\". (See: Strategy)
  • SubductionMethod => a string, default value "Top", the method used for subduction either: \"Top\" or \"Engine\". (See: SubductionMethod)
  • Limit => an integer, default value 100, a degree limit for the binomial S-pairs that are computed internally.
  • AutoSubduceOnPartialCompletion => a Boolean value, default value false, apply autosubduction to the sagbi generators the first time no new generators are added. Use this only if very few new sagbi generators are expected. (See: AutoSubduceOnPartialCompletion)
  • PrintLevel => an integer, default value 0, When this is greater than zero, information is printed about the progress of the computation (See: PrintLevel)
  • 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 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, 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

See also

• sagbi -- Compute a subalgebra basis (sagbi basis)
• 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
• Strategy -- an optional argument
• SubductionMethod -- Subduction method for the Sagbi algorithm
• Limit -- an optional argument
• AutoSubduceOnPartialCompletion -- Subduct sagbi generators at the end of the Sagbi algorithm
• PrintLevel -- Levels of information displayed during Sagbi algorithm
• Recompute -- Flag for restarting a sagbi computation
• RenewOptions -- Flag for reselecting the options for a sagbi computation