\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra lingrob.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package LGROBP LinGroebnerPackage}
<<package LGROBP LinGroebnerPackage>>=
)abbrev package LGROBP LinGroebnerPackage
++  Given a Groebner basis B with respect to the total degree ordering for
++  a zero-dimensional ideal I, compute
++  a Groebner basis with respect to the lexicographical ordering by using
++  linear algebra.
LinGroebnerPackage(lv,F) : C == T

 where
  Z      ==>  Integer
  lv     :    List Symbol
  F      :    GcdDomain

  DP     ==>  DirectProduct(#lv,NonNegativeInteger)
  DPoly  ==>  DistributedMultivariatePolynomial(lv,F)

  HDP    ==>  HomogeneousDirectProduct(#lv,NonNegativeInteger)
  HDPoly ==>  HomogeneousDistributedMultivariatePolynomial(lv,F)

  OV     ==>  OrderedVariableList(lv)
  NNI    ==>  NonNegativeInteger
  LVals  ==>  Record(gblist : List DPoly,gvlist : List Z)
  VF     ==>  Vector F
  VV     ==>  Vector NNI
  MF     ==>  Matrix F
  cLVars ==>  Record(glbase:List DPoly,glval:List Z)

  C == with

     linGenPos    :           List HDPoly      -> LVals
	++ linGenPos \undocumented
     groebgen     :           List DPoly       -> cLVars
	++ groebgen \undocumented
     totolex      :           List HDPoly      -> List DPoly
	++ totolex \undocumented
     minPol       : (List HDPoly,List HDPoly,OV) -> HDPoly
	++ minPol \undocumented
     minPol       :        (List HDPoly,OV)     -> HDPoly
	++ minPol \undocumented


     computeBasis :        List HDPoly           -> List HDPoly
	++ computeBasis \undocumented
     coord        : (HDPoly,List HDPoly)         -> VF
	++ coord \undocumented
     anticoord    : (List F,DPoly,List DPoly)    -> DPoly
	++ anticoord \undocumented
     intcompBasis : (OV,List HDPoly,List HDPoly) -> List HDPoly
	++ intcompBasis \undocumented
     choosemon    :     (DPoly,List  DPoly)      -> DPoly
	++ choosemon \undocumented
     transform    :           DPoly              -> HDPoly
	++ transform \undocumented


  T == add

    import GroebnerPackage(F,DP,OV,DPoly)
    import GroebnerPackage(F,HDP,OV,HDPoly)
    import GroebnerInternalPackage(F,HDP,OV,HDPoly)
    import GroebnerInternalPackage(F,DP,OV,DPoly)

    lvar :=[variable(yx)::OV for yx in lv]

    reduceRow(M:MF, v : VF, lastRow: Integer, pivots: Vector(Integer)) : VF ==
      a1:F := 1
      b:F := 0
      dim := #v
      for j in 1..lastRow repeat -- scan over rows
         mj := row(M,j)
         k:=pivots(j)
         b:=mj.k
         vk := v.k
         for kk in 1..(k-1) repeat
            v(kk) := ((-b*v(kk)) exquo a1) :: F
         for kk in k..dim repeat
            v(kk) := ((vk*mj(kk)-b*v(kk)) exquo a1)::F
         a1 := b
      v

    rRedPol(f:HDPoly, B:List HDPoly):Record(poly:HDPoly, mult:F) ==
      gm := redPo(f,B)
      gm.poly = 0 => gm
      gg := reductum(gm.poly)
      ggm := rRedPol(gg,B)
      [ggm.mult*(gm.poly - gg) + ggm.poly, ggm.mult*gm.mult]

----- transform the total basis B in lex basis -----
    totolex(B : List HDPoly) : List DPoly ==
      result:List DPoly :=[]
      ltresult:List DPoly :=[]
      vBasis:= computeBasis B
      nBasis:List DPoly :=[1$DPoly]
      ndim:=(#vBasis)::PositiveInteger
      ndim1:NNI:=ndim+1
      lm:VF
      linmat:MF:=zero(ndim,2*ndim+1)
      linmat(1,1):=1$F
      linmat(1,ndim1):=1
      pivots:Vector Integer := new(ndim,0)
      pivots(1) := 1
      firstmon:DPoly:=1$DPoly
      ofirstmon:DPoly:=1$DPoly
      orecfmon:Record(poly:HDPoly, mult:F) := [1,1]
      i:NNI:=2
      while not one?(firstmon:=choosemon(firstmon,ltresult)) repeat
        if (v:=firstmon exquo ofirstmon) case "failed" then
          recfmon:=rRedPol(transform firstmon,B)
        else
          recfmon:=rRedPol(transform(v::DPoly) *orecfmon.poly,B)
          recfmon.mult := recfmon.mult * orecfmon.mult
        cc := gcd(content recfmon.poly, recfmon.mult)
        recfmon.poly := (recfmon.poly exquo cc)::HDPoly
        recfmon.mult := (recfmon.mult exquo cc)::F
        veccoef:VF:=coord(recfmon.poly,vBasis)
        ofirstmon:=firstmon
        orecfmon := recfmon
        lm:=zero(2*ndim+1)
        for j in 1..ndim repeat lm(j):=veccoef(j)
        lm(ndim+i):=recfmon.mult
        lm := reduceRow(linmat, lm, i-1, pivots)
        j : Integer
        if i=ndim1 then j:=ndim1
        else
          j:=1
          while lm(j) = 0 and j< ndim1 repeat j:=j+1
        if j=ndim1 then
          cordlist:List F:=[lm(k) for k in ndim1..ndim1+(#nBasis)]
          antc:=+/[c*b for c in reverse cordlist
                       for b in concat(firstmon,nBasis)]
          antc:=primitivePart antc
          result:=concat(antc,result)
          ltresult:=concat(antc-reductum antc,ltresult)
        else
          pivots(i) := j
          setRow!(linmat,i,lm)
          i:=i+1
          nBasis:=cons(firstmon,nBasis)
      result

---- Compute the univariate polynomial for x
----oldBasis is a total degree Groebner basis
    minPol(oldBasis:List HDPoly,x:OV) :HDPoly ==
      algBasis:= computeBasis oldBasis
      minPol(oldBasis,algBasis,x)

---- Compute the univariate polynomial for x
---- oldBasis is total Groebner, algBasis is the basis as algebra
    minPol(oldBasis:List HDPoly,algBasis:List HDPoly,x:OV) :HDPoly ==
      nvp:HDPoly:=x::HDPoly
      f:=1$HDPoly
      omult:F :=1
      ndim:=(#algBasis)::PositiveInteger
      ndim1:NNI:=ndim+1
      lm:VF
      linmat:MF:=zero(ndim,2*ndim+1)
      linmat(1,1):=1$F
      linmat(1,ndim1):=1
      pivots:Vector Integer := new(ndim,0)
      pivots(1) := 1
      for i in 2..ndim1 repeat
        recf:=rRedPol(f*nvp,oldBasis)
        omult := recf.mult * omult
        f := recf.poly
        cc := gcd(content f, omult)
        f := (f exquo cc)::HDPoly
        omult := (omult exquo cc)::F
        veccoef:VF:=coord(f,algBasis)
        lm:=zero(2*ndim+1)
        for j in 1..ndim repeat lm(j) := veccoef(j)
        lm(ndim+i):=omult
        lm := reduceRow(linmat, lm, i-1, pivots)
        j: Integer :=1
        while lm(j)=0 and j<ndim1 repeat j:=j+1
        if j=ndim1 then return
          g:HDPoly:=0
          for k in ndim1..2*ndim+1 repeat
            g:=g+lm(k) * nvp**((k-ndim1):NNI)
          primitivePart g
        pivots(i) := j
        setRow!(linmat,i,lm)

----- transform a DPoly in a HDPoly -----
    transform(dpol:DPoly) : HDPoly ==
      dpol=0 => 0$HDPoly
      monomial(leadingCoefficient dpol,
               directProduct(degree(dpol)::VV)$HDP)$HDPoly +
                                      transform(reductum dpol)

----- compute the basis for the vector space determined by B -----
    computeBasis(B:List HDPoly) : List HDPoly ==
      mB:List HDPoly:=[monomial(1$F,degree f)$HDPoly for f in B]
      result:List HDPoly := [1$HDPoly]
      for var in lvar repeat
        part:=intcompBasis(var,result,mB)
        result:=concat(result,part)
      result

----- internal function for computeBasis -----
    intcompBasis(x:OV,lr:List HDPoly,mB : List HDPoly):List HDPoly ==
      lr=[] => lr
      part:List HDPoly :=[]
      for f in lr repeat
        g:=x::HDPoly * f
        if not zero?(redPo(g,mB).poly) then part:=concat(g,part)
      concat(part,intcompBasis(x,part,mB))

----- coordinate of f with respect to the basis B -----
----- f is a reduced polynomial -----
    coord(f:HDPoly,B:List HDPoly) : VF ==
      ndim := #B
      vv:VF:=new(ndim,0$F)$VF
      while not zero? f repeat
        rf := reductum f
        lf := f-rf
        lcf := leadingCoefficient f
        i:Z:=position(monomial(1$F,degree lf),B)
        vv.i:=lcf
        f := rf
      vv

----- reconstruct the polynomial from its coordinate -----
    anticoord(vv:List F,mf:DPoly,B:List DPoly) : DPoly ==
      for f in B for c in vv repeat (mf:=mf-c*f)
      mf

----- choose the next monom -----
    choosemon(mf:DPoly,nB:List DPoly) : DPoly ==
      nB = [] => ((lvar.last)::DPoly)*mf
      for x in reverse lvar repeat
        xx:=x ::DPoly
        mf:=xx*mf
        if redPo(mf,nB).poly ~= 0 then return mf
        dx := degree(mf,x)
        mf := (mf exquo (xx ** dx))::DPoly
      mf

----- put B in general position, B is Groebner -----
    linGenPos(B : List HDPoly) : LVals ==
      result:List DPoly :=[]
      ltresult:List DPoly :=[]
      vBasis:= computeBasis B
      nBasis:List DPoly :=[1$DPoly]
      ndim:=#vBasis : PositiveInteger
      ndim1:NNI:=ndim+1
      lm:VF
      linmat:MF:=zero(ndim,2*ndim+1)
      linmat(1,1):=1$F
      linmat(1,ndim1):=1
      pivots:Vector Integer := new(ndim,0)
      pivots(1) := 1
      i:NNI:=2
      rval:List Z :=[]
      for ii in 1..(#lvar-1) repeat
        c:Z:=0
        while c=0 repeat c:=random(11)$Z
        rval:=concat(c,rval)
      nval:DPoly := (last.lvar)::DPoly -
                (+/[r*(vv)::DPoly for r in rval for vv in lvar])
      firstmon:DPoly:=1$DPoly
      ofirstmon:DPoly:=1$DPoly
      orecfmon:Record(poly:HDPoly, mult:F) := [1,1]
      lx:= lvar.last
      while not one?(firstmon:=choosemon(firstmon,ltresult)) repeat
        if (v:=firstmon exquo ofirstmon) case "failed" then
          recfmon:=rRedPol(transform(eval(firstmon,lx,nval)),B)
        else
          recfmon:=rRedPol(transform(eval(v,lx,nval))*orecfmon.poly,B)
          recfmon.mult := recfmon.mult * orecfmon.mult
        cc := gcd(content recfmon.poly, recfmon.mult)
        recfmon.poly := (recfmon.poly exquo cc)::HDPoly
        recfmon.mult := (recfmon.mult exquo cc)::F
        veccoef:VF:=coord(recfmon.poly,vBasis)
        ofirstmon:=firstmon
        orecfmon := recfmon
        lm:=zero(2*ndim+1)
        for j in 1..ndim repeat lm(j):=veccoef(j)
        lm(ndim+i):=recfmon.mult
        lm := reduceRow(linmat, lm, i-1, pivots)
        j: Integer :=1
        while lm(j) = 0 and j<ndim1 repeat j:=j+1
        if j=ndim1 then
          cordlist:List F:=[lm(j) for j in ndim1..ndim1+(#nBasis)]
          antc:=+/[c*b for c in reverse cordlist
                       for b in concat(firstmon,nBasis)]
          result:=concat(primitivePart antc,result)
          ltresult:=concat(antc-reductum antc,ltresult)
        else
          pivots(i) := j
          setRow!(linmat,i,lm)
          i:=i+1
          nBasis:=concat(firstmon,nBasis)
      [result,rval]$LVals

----- given a basis of a zero-dimensional ideal,
----- performs a random change of coordinates
----- computes a Groebner basis for the lex ordering
    groebgen(L:List DPoly) : cLVars ==
      xn:=lvar.last
      val := xn::DPoly
      nvar1:NNI:=(#lvar-1):NNI
      ll: List Z :=[random(11)$Z for i in 1..nvar1]
      val:=val+ +/[ll.i*(lvar.i)::DPoly for i in 1..nvar1]
      LL:=[elt(univariate(f,xn),val) for f in L]
      LL:=  groebner(LL)
      [LL,ll]$cLVars

@
\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 LGROBP LinGroebnerPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}