path: root/src/algebra/groebf.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra groebf.spad}
\author{H. Michael Moeller, Johannes Grabmeier}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject

\section{package GBF GroebnerFactorizationPackage}

<<package GBF GroebnerFactorizationPackage>>=
import Boolean
import List
)abbrev package GBF GroebnerFactorizationPackage
++ Author: H. Michael Moeller, Johannes Grabmeier
++ Date Created: 24 August 1989
++ Date Last Updated: 01 January 1992
++ Basic Operations: groebnerFactorize factorGroebnerBasis
++ Related Constructors:
++ Also See: GroebnerPackage, Ideal, IdealDecompositionPackage
++ AMS Classifications:
++ Keywords: groebner basis, groebner factorization, ideal decomposition
++ References:
++ Description:
++   \spadtype{GroebnerFactorizationPackage} provides the function
++   groebnerFactor" which uses the factorization routines of \Language{} to
++   factor each polynomial under consideration while doing the groebner basis
++   algorithm. Then it writes the ideal as an intersection of ideals
++   determined by the irreducible factors. Note that the whole ring may
++   occur as well as other redundancies. We also use the fact, that from the
++   second factor on we can assume that the preceding factors are
++   not equal to 0 and we divide all polynomials under considerations
++   by the elements of this list of "nonZeroRestrictions".
++   The result is a list of groebner bases, whose union of solutions
++   of the corresponding systems of equations is the solution of
++   the system of equation corresponding to the input list.
++   The term ordering is determined by the polynomial type used.
++   Suggested types include
++   \spadtype{DistributedMultivariatePolynomial},
++   \spadtype{HomogeneousDistributedMultivariatePolynomial},
++   \spadtype{GeneralDistributedMultivariatePolynomial}.

GroebnerFactorizationPackage(Dom, Expon, VarSet, Dpol): T == C where

 Dom :    Join(EuclideanDomain,CharacteristicZero)
 Expon :  OrderedAbelianMonoidSup
 VarSet : OrderedSet
 Dpol: PolynomialCategory(Dom, Expon, VarSet)
 MF       ==>   MultivariateFactorize(VarSet,Expon,Dom,Dpol)
 sugarPol ==>   Record(totdeg: NonNegativeInteger, pol : Dpol)
 critPair ==>   Record(lcmfij: Expon,totdeg: NonNegativeInteger, poli: Dpol, polj: Dpol )
 L        ==>   List
 B        ==>   Boolean
 NNI      ==>   NonNegativeInteger
 OUT      ==>   OutputForm

 T ==> with

   factorGroebnerBasis : L Dpol -> L L Dpol
     ++ factorGroebnerBasis(basis) checks whether the basis contains
     ++ reducible polynomials and uses these to split the basis.
   factorGroebnerBasis : (L Dpol, Boolean) -> L L Dpol
     ++ factorGroebnerBasis(basis,info) checks whether the basis contains
     ++ reducible polynomials and uses these to split the basis.
     ++ If argument {\em info} is true, information is printed about
     ++ partial results.
   groebnerFactorize : (L Dpol, L Dpol) -> L L Dpol
     ++ groebnerFactorize(listOfPolys, nonZeroRestrictions) returns
     ++ a list of groebner basis. The union of their solutions
     ++ is the solution of the system of equations given by {\em listOfPolys}
     ++ under the restriction that the polynomials of {\em nonZeroRestrictions}
     ++ don't vanish.
     ++ At each stage the polynomial p under consideration (either from
     ++ the given basis or obtained from a reduction of the next S-polynomial)
     ++ is factorized. For each irreducible factors of p, a
     ++ new {\em createGroebnerBasis} is started
     ++ doing the usual updates with the factor
     ++ in place of p.
   groebnerFactorize : (L Dpol, L Dpol, Boolean) -> L L Dpol
     ++ groebnerFactorize(listOfPolys, nonZeroRestrictions, info) returns
     ++ a list of groebner basis. The union of their solutions
     ++ is the solution of the system of equations given by {\em listOfPolys}
     ++ under the restriction that the polynomials of {\em nonZeroRestrictions}
     ++ don't vanish.
     ++ At each stage the polynomial p under consideration (either from
     ++ the given basis or obtained from a reduction of the next S-polynomial)
     ++ is factorized. For each irreducible factors of p a
     ++ new {\em createGroebnerBasis} is started 
     ++ doing the usual updates with the factor in place of p.
     ++ If argument {\em info} is true, information is printed about
     ++ partial results.
   groebnerFactorize : L Dpol  -> L L Dpol
     ++ groebnerFactorize(listOfPolys) returns 
     ++ a list of groebner bases. The union of their solutions
     ++ is the solution of the system of equations given by {\em listOfPolys}.
     ++ At each stage the polynomial p under consideration (either from
     ++ the given basis or obtained from a reduction of the next S-polynomial)
     ++ is factorized. For each irreducible factors of p, a
     ++ new {\em createGroebnerBasis} is started 
     ++ doing the usual updates with the factor 
     ++ in place of p.
   groebnerFactorize : (L Dpol, Boolean)  -> L L Dpol
     ++ groebnerFactorize(listOfPolys, info) returns
     ++ a list of groebner bases. The union of their solutions
     ++ is the solution of the system of equations given by {\em listOfPolys}.
     ++ At each stage the polynomial p under consideration (either from
     ++ the given basis or obtained from a reduction of the next S-polynomial)
     ++ is factorized. For each irreducible factors of p, a
     ++ new {\em createGroebnerBasis} is started 
     ++ doing the usual updates with the factor
     ++ in place of p.
     ++ If {\em info} is true, information is printed about partial results.

 C ==> add

   import GroebnerInternalPackage(Dom,Expon,VarSet,Dpol)
   -- next to help compiler to choose correct signatures:
   info: Boolean
   -- signatures of local functions

   newPairs : (L sugarPol, Dpol) -> L critPair
     -- newPairs(lp, p) constructs list of critical pairs from the list of
     -- {\em lp} of input polynomials and a given further one p.
     -- It uses criteria M and T to reduce the list.
   updateCritPairs : (L critPair, L critPair, Dpol) -> L critPair
     -- updateCritPairs(lcP1,lcP2,p) applies criterion B to {\em lcP1} using
     -- p. Then this list is merged with {\em lcP2}.
   updateBasis : (L sugarPol, Dpol, NNI) -> L sugarPol
     -- updateBasis(li,p,deg) every polynomial in {\em li} is dropped if
     -- its leading term is a multiple of the leading term of p.
     -- The result is this list enlarged by p.
   createGroebnerBases : (L sugarPol, L Dpol, L Dpol, L Dpol, L critPair,_
                          L L Dpol, Boolean) -> L L Dpol
     -- createGroebnerBases(basis, redPols, nonZeroRestrictions, inputPolys,
     --   lcP,listOfBases): This function is used to be called from
     -- groebnerFactorize.
     -- basis: part of a Groebner basis, computed so far
     -- redPols: Polynomials from the ideal to be used for reducing,
     --   we don't throw away polynomials
     -- nonZeroRestrictions: polynomials not zero in the common zeros
     --   of the polynomials in the final (Groebner) basis
     -- inputPolys: assumed to be in descending order
     -- lcP: list of critical pairs built from polynomials of the
     --   actual basis
     -- listOfBases: Collects the (Groebner) bases constructed by this
     --   recursive algorithm at different stages.
     --   we print info messages if info is true
   createAllFactors: Dpol -> L Dpol
     -- factor reduced critpair polynomial

   -- implementation of local functions


   createGroebnerBases(basis, redPols, nonZeroRestrictions, inputPolys,_
       lcP, listOfBases, info) ==
     doSplitting? : B := false
     terminateWithBasis : B := false
     allReducedFactors : L Dpol := []
     nP : Dpol  -- actual polynomial under consideration
     p :  Dpol  -- next polynomial from input list
     h :  Dpol  -- next polynomial from critical pairs
     stopDividing : Boolean
     --    STEP 1   do the next polynomials until a splitting is possible
     -- In the first step we take the first polynomial of "inputPolys"
     -- if empty, from list of critical pairs "lcP" and do the following:
     -- Divide it, if possible, by the polynomials from "nonZeroRestrictions".
     -- We factorize it and reduce each irreducible  factor with respect to
     -- "basis". If 0$Dpol occurs in the list we update the list and continue
     -- with next polynomial.
     -- If there are at least two (irreducible) factors
     -- in the list of factors we finish STEP 1 and set a boolean variable
     -- to continue with STEP 2, the splitting step.
     -- If there is just one of it, we do the following:
     -- If it is 1$Dpol we stop the whole calculation and put
     -- [1$Dpol] into the listOfBases
     -- Otherwise we update the "basis" and the other lists and continue
     -- with next polynomial.

     while (not doSplitting?) and (not terminateWithBasis) repeat
       terminateWithBasis := (null inputPolys and null lcP)
       not terminateWithBasis =>  -- still polynomials left
         -- determine next polynomial "nP"
         nP :=
           not null inputPolys =>
             p := first inputPolys
             inputPolys := rest inputPolys
             -- we know that p is not equal to 0 or 1, but, although,
             -- the inputPolys and the basis are ordered, we cannot assume
             -- that p is reduced w.r.t. basis, as the ordering is only quasi
             -- and we could have equal leading terms, and due to factorization
             -- polynomials of smaller leading terms, hence reduce p first:
             hMonic redPol(p,redPols)
           -- now we have inputPolys empty and hence lcP is not empty:
           -- create S-Polynomial from first critical pair:
           h := sPol first lcP
           lcP := rest lcP
           hMonic redPol(h,redPols)

         nP = 1$Dpol =>
           basis := [[0,1$Dpol]$sugarPol]
           terminateWithBasis := true

         -- if "nP" ~= 0, then  we continue, otherwise we determine next "nP"
         nP ~= 0$Dpol =>
           -- now we divide "nP", if possible, by the polynomials
           -- from "nonZeroRestrictions"
           for q in nonZeroRestrictions repeat
             stopDividing := false
             until stopDividing repeat
               nPq := nP exquo q
               if nPq case Dpol then
                 nP := nPq@Dpol
               else
                 stopDividing := true
               stopDividing := stopDividing or zero? degree nP

           zero? degree nP =>
             basis := [[0,1$Dpol]$sugarPol]
             terminateWithBasis := true  -- doSplitting? is still false

           -- a careful analysis has to be done, when and whether the
           -- following reduction and case nP=1 is necessary

           nP := hMonic redPol(nP,redPols)
           zero? degree nP =>
             basis := [[0,1$Dpol]$sugarPol]
             terminateWithBasis := true  -- doSplitting? is still false

           -- if "nP" ~= 0, then  we continue, otherwise we determine next "nP"
           nP ~= 0$Dpol =>
             -- now we factorize "nP", which is not constant
             irreducibleFactors : L Dpol := createAllFactors(nP)
             -- if there are more than 1 factors we reduce them and split
             (doSplitting? := not null rest irreducibleFactors) =>
               -- and reduce and normalize the factors
               for fnP in irreducibleFactors repeat
                 fnP := hMonic redPol(fnP,redPols)
                 -- no factor reduces to 0, as then "fP" would have been
                 -- reduced to zero,
                 -- but 1 may occur, which we will drop in a later version.
                 allReducedFactors := cons(fnP, allReducedFactors)
               -- end of "for fnP in irreducibleFactors repeat"

               -- we want that the smaller factors are dealt with first
               allReducedFactors := reverse allReducedFactors
             -- now the case of exactly 1 factor, but certainly not
             -- further reducible with respect to "redPols"
             nP := first irreducibleFactors
             -- put "nP" into "basis" and update "lcP" and "redPols":
             lcP : L critPair := updateCritPairs(lcP,newPairs(basis,nP),nP)
             basis := updateBasis(basis,nP,virtualDegree nP)
             redPols := concat(redPols,nP)
     -- end of "while not doSplitting? and not terminateWithBasis repeat"

     --    STEP 2  splitting step

     doSplitting? =>
       for fnP in allReducedFactors repeat
         if not one? fnP
           then
             newInputPolys : L Dpol  := _
               sort( degree #1 > degree #2 ,cons(fnP,inputPolys))
             listOfBases := createGroebnerBases(basis, redPols, _
               nonZeroRestrictions,newInputPolys,lcP,listOfBases,info)
             -- update "nonZeroRestrictions"
             nonZeroRestrictions := cons(fnP,nonZeroRestrictions)
           else
             if info then
               messagePrint("we terminated with [1]")$OUT
             listOfBases := cons([1$Dpol],listOfBases)

       -- we finished with all the branches on one level and hence
       -- finished this call of createGroebnerBasis. Therefore
       -- we terminate with the actual "listOfBasis" as
       -- everything is done in the recursions
       listOfBases
     -- end of "doSplitting? =>"

     --    STEP 3 termination step

     --  we found a groebner basis and put it into the list "listOfBases"
     --  (auto)reduce each basis element modulo the others
     newBasis := minGbasis(sort(degree #1 > degree #2,[p.pol for p in basis]))
     -- now check whether the normalized basis again has reducible
     -- polynomials, in this case continue splitting!
     if info then
       messagePrint("we found a groebner basis and check whether it ")$OUT
       messagePrint("contains reducible polynomials")$OUT
       print(newBasis::OUT)$OUT
       -- here we should create an output form which is reusable by the system
       -- print(convert(newBasis::OUT)$InputForm :: OUT)$OUT
     removeDuplicates append(factorGroebnerBasis(newBasis, info), listOfBases)

   createAllFactors(p: Dpol) ==
     loF : L Dpol := [el.fctr for el in factorList factor(p)$MF]
     sort(degree #1 < degree #2, loF)
   newPairs(lp : L sugarPol,p : Dpol) ==
     totdegreeOfp : NNI := virtualDegree p
     -- next list lcP contains all critPair constructed from
     -- p and and the polynomials q in lp
     lcP: L critPair := _
       --[[sup(degree q, degreeOfp), q, p]$critPair for q in lp]
       [makeCrit(q, p, totdegreeOfp) for q in lp]
     -- application of the criteria to reduce the list lcP
     critMTonD1 sort(critpOrder,lcP)
   updateCritPairs(oldListOfcritPairs, newListOfcritPairs, p)==
     updatD (newListOfcritPairs, critBonD(p,oldListOfcritPairs))
   updateBasis(lp, p, deg) == updatF(p,deg,lp)

   -- exported functions

   factorGroebnerBasis basis == factorGroebnerBasis(basis, false)

   factorGroebnerBasis (basis, info) ==
     foundAReducible : Boolean := false
     for p in basis while not foundAReducible repeat
       -- we use fact that polynomials have content 1
       foundAReducible := 1 < #[el.fctr for el in factorList factor(p)$MF]
     not foundAReducible =>
       if info then  messagePrint("factorGroebnerBasis: no reducible polynomials in this basis")$OUT
       [basis]
     -- improve! Use the fact that the irreducible ones already
     -- build part of the basis, use the done factorizations, etc.
     if info then  messagePrint("factorGroebnerBasis:_
        we found reducible polynomials and continue splitting")$OUT
     createGroebnerBases([],[],[],basis,[],[],info)

   groebnerFactorize(basis: List Dpol, nonZeroRestrictions: List Dpol) ==
     groebnerFactorize(basis, nonZeroRestrictions, false)

   groebnerFactorize(basis, nonZeroRestrictions, info) ==
     basis = [] => [basis]
     basis := remove(#1 = 0$Dpol,basis)
     basis = [] => [[0$Dpol]]
     -- normalize all input polynomial
     basis := [hMonic p for p in basis]
     member?(1$Dpol,basis) => [[1$Dpol]]
     basis :=  sort(degree #1 > degree #2, basis)
     createGroebnerBases([],[],nonZeroRestrictions,basis,[],[],info)

   groebnerFactorize(basis) == groebnerFactorize(basis, [], false)
   groebnerFactorize(basis: List Dpol,info: Boolean) == 
     groebnerFactorize(basis, [], info)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package GBF GroebnerFactorizationPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}