/kernel/ideals.cc

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/****************************************
*  Computer Algebra System SINGULAR     *
****************************************/
/* $Id$ */
/*
* ABSTRACT - all basic methods to manipulate ideals
*/

/* includes */
#include <kernel/mod2.h>

#ifndef NDEBUG
# define MYTEST 0
#else /* ifndef NDEBUG */
# define MYTEST 1
#endif /* ifndef NDEBUG */

#include <kernel/options.h>
#include <omalloc/omalloc.h>
#include <kernel/febase.h>
#include <kernel/numbers.h>
#include <kernel/longrat.h>
#include <kernel/polys.h>
#include <kernel/ring.h>
#include <kernel/kstd1.h>
#include <kernel/matpol.h>
#include <kernel/weight.h>
#include <kernel/intvec.h>
#include <kernel/syz.h>
#include <kernel/sparsmat.h>
#include <kernel/ideals.h>
#include <kernel/prCopy.h>
#include <kernel/gring.h>


omBin sip_sideal_bin = omGetSpecBin(sizeof(sip_sideal));

/* #define WITH_OLD_MINOR */
#define pCopy_noCheck(p) pCopy(p)

static poly * idpower;
/*collects the monomials in makemonoms, must be allocated befor*/
static int idpowerpoint;
/*index of the actual monomial in idpower*/
static poly * givenideal;
/*the ideal from which a power is computed*/

/*0 implementation*/

/*2
* initialise an ideal
*/
ideal idInit(int idsize, int rank)
{
  /*- initialise an ideal -*/
  ideal hh = (ideal )omAllocBin(sip_sideal_bin);
  hh->nrows = 1;
  hh->rank = rank;
  IDELEMS(hh) = idsize;
  if (idsize>0)
  {
    hh->m = (poly *)omAlloc0(idsize*sizeof(poly));
  }
  else
    hh->m=NULL;
  return hh;
}

#ifdef PDEBUG
// this is only for outputting an ideal within the debugger
void idShow(const ideal id, const ring lmRing, const ring tailRing, const int debugPrint)
{
  assume( debugPrint >= 0 );

  if( id == NULL )
    PrintS("(NULL)");
  else
  {
    Print("Module of rank %ld,real rank %ld and %d generators.\n",
          id->rank,idRankFreeModule(id, lmRing, tailRing),IDELEMS(id));

    int j = (id->ncols*id->nrows) - 1;
    while ((j > 0) && (id->m[j]==NULL)) j--;
    for (int i = 0; i <= j; i++)
    {
      Print("generator %d: ",i); p_DebugPrint(id->m[i], lmRing, tailRing, debugPrint);
    }
  }
}
#endif

/* index of generator with leading term in ground ring (if any);
   otherwise -1 */
int idPosConstant(ideal id)
{
  int k;
  for (k = IDELEMS(id)-1; k>=0; k--)
  {
    if (p_LmIsConstantComp(id->m[k], currRing) == TRUE)
      return k;
  }
  return -1;
}

/*2
* initialise the maximal ideal (at 0)
*/
ideal idMaxIdeal (void)
{
  int l;
  ideal hh=NULL;

  hh=idInit(pVariables,1);
  for (l=0; l<pVariables; l++)
  {
    hh->m[l] = pOne();
    pSetExp(hh->m[l],l+1,1);
    pSetm(hh->m[l]);
  }
  return hh;
}

/*2
* deletes an ideal/matrix
*/
void id_Delete (ideal * h, ring r)
{
  int j,elems;
  if (*h == NULL)
    return;
  elems=j=(*h)->nrows*(*h)->ncols;
  if (j>0)
  {
    do
    {
      p_Delete(&((*h)->m[--j]), r);
    }
    while (j>0);
    omFreeSize((ADDRESS)((*h)->m),sizeof(poly)*elems);
  }
  omFreeBin((ADDRESS)*h, sip_sideal_bin);
  *h=NULL;
}


/*2
* Shallowdeletes an ideal/matrix
*/
void id_ShallowDelete (ideal *h, ring r)
{
  int j,elems;
  if (*h == NULL)
    return;
  elems=j=(*h)->nrows*(*h)->ncols;
  if (j>0)
  {
    do
    {
      p_ShallowDelete(&((*h)->m[--j]), r);
    }
    while (j>0);
    omFreeSize((ADDRESS)((*h)->m),sizeof(poly)*elems);
  }
  omFreeBin((ADDRESS)*h, sip_sideal_bin);
  *h=NULL;
}

/*2
*gives an ideal the minimal possible size
*/
void idSkipZeroes (ideal ide)
{
  int k;
  int j = -1;
  BOOLEAN change=FALSE;
  for (k=0; k<IDELEMS(ide); k++)
  {
    if (ide->m[k] != NULL)
    {
      j++;
      if (change)
      {
        ide->m[j] = ide->m[k];
      }
    }
    else
    {
      change=TRUE;
    }
  }
  if (change)
  {
    if (j == -1)
      j = 0;
    else
    {
      for (k=j+1; k<IDELEMS(ide); k++)
        ide->m[k] = NULL;
    }
    pEnlargeSet(&(ide->m),IDELEMS(ide),j+1-IDELEMS(ide));
    IDELEMS(ide) = j+1;
  }
}

/*2
* copies the first k (>= 1) entries of the given ideal
* and returns these as a new ideal
* (Note that the copied polynomials may be zero.)
*/
ideal idCopyFirstK (const ideal ide, const int k)
{
  ideal newI = idInit(k, 0);
  for (int i = 0; i < k; i++)
    newI->m[i] = pCopy(ide->m[i]);
  return newI;
}

/*2
* ideal id = (id[i])
* result is leadcoeff(id[i]) = 1
*/
void idNorm(ideal id)
{
  for (int i=IDELEMS(id)-1; i>=0; i--)
  {
    if (id->m[i] != NULL)
    {
      pNorm(id->m[i]);
    }
  }
}

/*2
* ideal id = (id[i]), c any unit
* if id[i] = c*id[j] then id[j] is deleted for j > i
*/
void idDelMultiples(ideal id)
{
  int i, j;
  int k = IDELEMS(id)-1;
  for (i=k; i>=0; i--)
  {
    if (id->m[i]!=NULL)
    {
      for (j=k; j>i; j--)
      {
        if (id->m[j]!=NULL)
        {
#ifdef HAVE_RINGS
          if (rField_is_Ring(currRing))
          {
            /* if id[j] = c*id[i] then delete id[j].
               In the below cases of a ground field, we
               check whether id[i] = c*id[j] and, if so,
               delete id[j] for historical reasons (so
               that previous output does not change) */
            if (pComparePolys(id->m[j], id->m[i])) pDelete(&id->m[j]);
          }
          else
          {
            if (pComparePolys(id->m[i], id->m[j])) pDelete(&id->m[j]);
          }
#else
          if (pComparePolys(id->m[i], id->m[j])) pDelete(&id->m[j]);
#endif          
        }
      }
    }
  }
}

/*2
* ideal id = (id[i])
* if id[i] = id[j] then id[j] is deleted for j > i
*/
void idDelEquals(ideal id)
{
  int i, j;
  int k = IDELEMS(id)-1;
  for (i=k; i>=0; i--)
  {
    if (id->m[i]!=NULL)
    {
      for (j=k; j>i; j--)
      {
        if ((id->m[j]!=NULL)
        && (pEqualPolys(id->m[i], id->m[j])))
        {
          pDelete(&id->m[j]);
        }
      }
    }
  }
}

//
// Delete id[j], if Lm(j) == Lm(i) and both LC(j), LC(i) are units and j > i
//
void idDelLmEquals(ideal id)
{
  int i, j;
  int k = IDELEMS(id)-1;
  for (i=k; i>=0; i--)
  {
    if (id->m[i] != NULL)
    {
      for (j=k; j>i; j--)
      {
        if ((id->m[j] != NULL)
        && pLmEqual(id->m[i], id->m[j])
#ifdef HAVE_RINGS
        && nIsUnit(pGetCoeff(id->m[i])) && nIsUnit(pGetCoeff(id->m[j]))
#endif
        )
        {
          pDelete(&id->m[j]);
        }
      }
    }
  }
}

//
// delete id[j], if LT(j) == coeff*mon*LT(i) and vice versa, i.e.,
// delete id[i], if LT(i) == coeff*mon*LT(j)
//
void idDelDiv(ideal id)
{
  int i, j;
  int k = IDELEMS(id)-1;
  for (i=k; i>=0; i--)
  {
    if (id->m[i] != NULL)
    {
      for (j=k; j>i; j--)
      {
        if (id->m[j]!=NULL)
        {
#ifdef HAVE_RINGS
          if (rField_is_Ring(currRing))
          {
            if (pDivisibleByRingCase(id->m[i], id->m[j]))
            {
              pDelete(&id->m[j]);
            }
            else if (pDivisibleByRingCase(id->m[j], id->m[i]))
          {
            pDelete(&id->m[i]);
            break;
          }
          }
          else
          {
#endif
          /* the case of a ground field: */
          if (pDivisibleBy(id->m[i], id->m[j]))
          {
            pDelete(&id->m[j]);
          }
          else if (pDivisibleBy(id->m[j], id->m[i]))
          {
            pDelete(&id->m[i]);
            break;
          }
#ifdef HAVE_RINGS
          }
#endif          
        }
      }
    }
  }
}

/*2
*test if the ideal has only constant polynomials
*/
BOOLEAN idIsConstant(ideal id)
{
  int k;
  for (k = IDELEMS(id)-1; k>=0; k--)
  {
    if (pIsConstantPoly(id->m[k]) == FALSE)
      return FALSE;
  }
  return TRUE;
}

/*2
* copy an ideal
*/

ideal id_Copy (ideal h1, const ring r)
{
  int i;
  ideal h2;

//#ifdef TEST
  if (h1 == NULL)
  {
    h2=idInit(1,1);
  }
  else
//#endif
  {
    h2=idInit(IDELEMS(h1),h1->rank);
    for (i=IDELEMS(h1)-1; i>=0; i--)
      h2->m[i] = p_Copy(h1->m[i],r);
  }
  return h2;
}

#ifdef PDEBUG
void idDBTest(ideal h1, int level, const char *f,const int l)
{
  int i;

  if (h1 != NULL)
  {
    // assume(IDELEMS(h1) > 0); for ideal/module, does not apply to matrix
    omCheckAddrSize(h1,sizeof(*h1));
    omdebugAddrSize(h1->m,h1->ncols*h1->nrows*sizeof(poly));
    /* to be able to test matrices: */
    for (i=(h1->ncols*h1->nrows)-1; i>=0; i--)
      _p_Test(h1->m[i], currRing, level);
    int new_rk=idRankFreeModule(h1);
    if(new_rk > h1->rank)
    {
      dReportError("wrong rank %d (should be %d) in %s:%d\n",
                   h1->rank, new_rk, f,l);
      omPrintAddrInfo(stderr, h1, " for ideal");
      h1->rank=new_rk;
    }
  }
}
#endif

/*3
* for idSort: compare a and b revlex inclusive module comp.
*/
static int pComp_RevLex(poly a, poly b,BOOLEAN nolex)
{
  if (b==NULL) return 1;
  if (a==NULL) return -1;

  if (nolex) 
  {
    int r=pLmCmp(a,b);
    if (r!=0) return r;
    number h=nSub(pGetCoeff(a),pGetCoeff(b));
    r = -1+nIsZero(h)+2*nGreaterZero(h); /* -1: <, 0:==, 1: > */
    nDelete(&h);
    return r;
  }
  int l=pVariables;
  while ((l>0) && (pGetExp(a,l)==pGetExp(b,l))) l--;
  if (l==0)
  {
    if (pGetComp(a)==pGetComp(b))
    {
      number h=nSub(pGetCoeff(a),pGetCoeff(b));
      int r = -1+nIsZero(h)+2*nGreaterZero(h); /* -1: <, 0:==, 1: > */
      nDelete(&h);
      return r;
    }
    if (pGetComp(a)>pGetComp(b)) return 1;
  }
  else if (pGetExp(a,l)>pGetExp(b,l))
    return 1;
  return -1;
}

/*2
*sorts the ideal w.r.t. the actual ringordering
*uses lex-ordering when nolex = FALSE
*/
intvec *idSort(ideal id,BOOLEAN nolex)
{
  poly p,q;
  intvec * result = new intvec(IDELEMS(id));
  int i, j, actpos=0, newpos, l;
  int diff, olddiff, lastcomp, newcomp;
  BOOLEAN notFound;

  for (i=0;i<IDELEMS(id);i++)
  {
    if (id->m[i]!=NULL)
    {
      notFound = TRUE;
      newpos = actpos / 2;
      diff = (actpos+1) / 2;
      diff = (diff+1) / 2;
      lastcomp = pComp_RevLex(id->m[i],id->m[(*result)[newpos]],nolex);
      if (lastcomp<0)
      {
        newpos -= diff;
      }
      else if (lastcomp>0)
      {
        newpos += diff;
      }
      else
      {
        notFound = FALSE;
      }
      //while ((newpos>=0) && (newpos<actpos) && (notFound))
      while (notFound && (newpos>=0) && (newpos<actpos))
      {
        newcomp = pComp_RevLex(id->m[i],id->m[(*result)[newpos]],nolex);
        olddiff = diff;
        if (diff>1)
        {
          diff = (diff+1) / 2;
          if ((newcomp==1)
          && (actpos-newpos>1)
          && (diff>1)
          && (newpos+diff>=actpos))
          {
            diff = actpos-newpos-1;
          }
          else if ((newcomp==-1)
          && (diff>1)
          && (newpos<diff))
          {
            diff = newpos;
          }
        }
        if (newcomp<0)
        {
          if ((olddiff==1) && (lastcomp>0))
            notFound = FALSE;
          else
            newpos -= diff;
        }
        else if (newcomp>0)
        {
          if ((olddiff==1) && (lastcomp<0))
          {
            notFound = FALSE;
            newpos++;
          }
          else
          {
            newpos += diff;
          }
        }
        else
        {
          notFound = FALSE;
        }
        lastcomp = newcomp;
        if (diff==0) notFound=FALSE; /*hs*/
      }
      if (newpos<0) newpos = 0;
      if (newpos>actpos) newpos = actpos;
      while ((newpos<actpos) && (pComp_RevLex(id->m[i],id->m[(*result)[newpos]],nolex)==0))
        newpos++;
      for (j=actpos;j>newpos;j--)
      {
        (*result)[j] = (*result)[j-1];
      }
      (*result)[newpos] = i;
      actpos++;
    }
  }
  for (j=0;j<actpos;j++) (*result)[j]++;
  return result;
}

/*2
* concat the lists h1 and h2 without zeros
*/
ideal idSimpleAdd (ideal h1,ideal h2)
{
  int i,j,r,l;
  ideal result;

  if (h1==NULL) return idCopy(h2);
  if (h2==NULL) return idCopy(h1);
  j = IDELEMS(h1)-1;
  while ((j >= 0) && (h1->m[j] == NULL)) j--;
  i = IDELEMS(h2)-1;
  while ((i >= 0) && (h2->m[i] == NULL)) i--;
  r = si_max(h1->rank,h2->rank);
  if (i+j==(-2))
    return idInit(1,r);
  else
    result=idInit(i+j+2,r);
  for (l=j; l>=0; l--)
  {
    result->m[l] = pCopy(h1->m[l]);
  }
  r = i+j+1;
  for (l=i; l>=0; l--, r--)
  {
    result->m[r] = pCopy(h2->m[l]);
  }
  return result;
}

/*2
* insert h2 into h1 (if h2 is not the zero polynomial)
* return TRUE iff h2 was indeed inserted
*/
BOOLEAN idInsertPoly (ideal h1, poly h2)
{
  if (h2==NULL) return FALSE;
  int j = IDELEMS(h1)-1;
  while ((j >= 0) && (h1->m[j] == NULL)) j--;
  j++;
  if (j==IDELEMS(h1))
  {
    pEnlargeSet(&(h1->m),IDELEMS(h1),16);
    IDELEMS(h1)+=16;
  }
  h1->m[j]=h2;
  return TRUE;
}

/*2
* insert h2 into h1 depending on the two boolean parameters:
* - if zeroOk is true, then h2 will also be inserted when it is zero
* - if duplicateOk is true, then h2 will also be inserted when it is
*   already present in h1
* return TRUE iff h2 was indeed inserted
*/
BOOLEAN idInsertPolyWithTests (ideal h1, const int validEntries,
  const poly h2, const bool zeroOk, const bool duplicateOk)
{
  if ((!zeroOk) && (h2 == NULL)) return FALSE;
  if (!duplicateOk)
  {
    bool h2FoundInH1 = false;
    int i = 0;
    while ((i < validEntries) && (!h2FoundInH1))
    {
      h2FoundInH1 = pEqualPolys(h1->m[i], h2);
      i++;
    }
    if (h2FoundInH1) return FALSE;
  }
  if (validEntries == IDELEMS(h1))
  {
    pEnlargeSet(&(h1->m), IDELEMS(h1), 16);
    IDELEMS(h1) += 16;
  }
  h1->m[validEntries] = h2;
  return TRUE;
}

/*2
* h1 + h2
*/
ideal idAdd (ideal h1,ideal h2)
{
  ideal result = idSimpleAdd(h1,h2);
  idCompactify(result);
  return result;
}

/*2
* h1 * h2
*/
ideal  idMult (ideal h1,ideal  h2)
{
  int i,j,k;
  ideal  hh;

  j = IDELEMS(h1);
  while ((j > 0) && (h1->m[j-1] == NULL)) j--;
  i = IDELEMS(h2);
  while ((i > 0) && (h2->m[i-1] == NULL)) i--;
  j = j * i;
  if (j == 0)
    hh = idInit(1,1);
  else
    hh=idInit(j,1);
  if (h1->rank<h2->rank)
    hh->rank = h2->rank;
  else
    hh->rank = h1->rank;
  if (j==0) return hh;
  k = 0;
  for (i=0; i<IDELEMS(h1); i++)
  {
    if (h1->m[i] != NULL)
    {
      for (j=0; j<IDELEMS(h2); j++)
      {
        if (h2->m[j] != NULL)
        {
          hh->m[k] = ppMult_qq(h1->m[i],h2->m[j]);
          k++;
        }
      }
    }
  }
  {
    idCompactify(hh);
    return hh;
  }
}

/*2
*returns true if h is the zero ideal
*/
BOOLEAN idIs0 (ideal h)
{
  int i;

  if (h == NULL) return TRUE;
  i = IDELEMS(h)-1;
  while ((i >= 0) && (h->m[i] == NULL))
  {
    i--;
  }
  if (i < 0)
    return TRUE;
  else
    return FALSE;
}

/*2
* return the maximal component number found in any polynomial in s
*/
long idRankFreeModule (ideal s, ring lmRing, ring tailRing)
{
  if (s!=NULL)
  {
    int  j=0;

    if (rRing_has_Comp(tailRing) && rRing_has_Comp(lmRing))
    {
      int  l=IDELEMS(s);
      poly *p=s->m;
      int  k;
      for (; l != 0; l--)
      {
        if (*p!=NULL)
        {
          pp_Test(*p, lmRing, tailRing);
          k = p_MaxComp(*p, lmRing, tailRing);
          if (k>j) j = k;
        }
        p++;
      }
    }
    return j;
  }
  return -1;
}

BOOLEAN idIsModule(ideal id, ring r)
{
  if (id != NULL && rRing_has_Comp(r))
  {
    int j, l = IDELEMS(id);
    for (j=0; j<l; j++)
    {
      if (id->m[j] != NULL && p_GetComp(id->m[j], r) > 0) return TRUE;
    }
  }
  return FALSE;
}


/*2
*returns true if id is homogenous with respect to the aktual weights
*/
BOOLEAN idHomIdeal (ideal id, ideal Q)
{
  int i;
  BOOLEAN     b;
  if ((id == NULL) || (IDELEMS(id) == 0)) return TRUE;
  i = 0;
  b = TRUE;
  while ((i < IDELEMS(id)) && b)
  {
    b = pIsHomogeneous(id->m[i]);
    i++;
  }
  if ((b) && (Q!=NULL) && (IDELEMS(Q)>0))
  {
    i=0;
    while ((i < IDELEMS(Q)) && b)
    {
      b = pIsHomogeneous(Q->m[i]);
      i++;
    }
  }
  return b;
}

/*2
*returns a minimized set of generators of h1
*/
ideal idMinBase (ideal h1)
{
  ideal h2, h3,h4,e;
  int j,k;
  int i,l,ll;
  intvec * wth;
  BOOLEAN homog;

  homog = idHomModule(h1,currQuotient,&wth);
  if (rHasGlobalOrdering_currRing())
  {
    if(!homog)
    {
      WarnS("minbase applies only to the local or homogeneous case");
      e=idCopy(h1);
      return e;
    }
    else
    {
      ideal re=kMin_std(h1,currQuotient,(tHomog)homog,&wth,h2,NULL,0,3);
      idDelete(&re);
      return h2;
    }
  }
  e=idInit(1,h1->rank);
  if (idIs0(h1))
  {
    return e;
  }
  pEnlargeSet(&(e->m),IDELEMS(e),15);
  IDELEMS(e) = 16;
  h2 = kStd(h1,currQuotient,isNotHomog,NULL);
  h3 = idMaxIdeal();
  h4=idMult(h2,h3);
  idDelete(&h3);
  h3=kStd(h4,currQuotient,isNotHomog,NULL);
  k = IDELEMS(h3);
  while ((k > 0) && (h3->m[k-1] == NULL)) k--;
  j = -1;
  l = IDELEMS(h2);
  while ((l > 0) && (h2->m[l-1] == NULL)) l--;
  for (i=l-1; i>=0; i--)
  {
    if (h2->m[i] != NULL)
    {
      ll = 0;
      while ((ll < k) && ((h3->m[ll] == NULL)
      || !pDivisibleBy(h3->m[ll],h2->m[i])))
        ll++;
      if (ll >= k)
      {
        j++;
        if (j > IDELEMS(e)-1)
        {
          pEnlargeSet(&(e->m),IDELEMS(e),16);
          IDELEMS(e) += 16;
        }
        e->m[j] = pCopy(h2->m[i]);
      }
    }
  }
  idDelete(&h2);
  idDelete(&h3);
  idDelete(&h4);
  if (currQuotient!=NULL)
  {
    h3=idInit(1,e->rank);
    h2=kNF(h3,currQuotient,e);
    idDelete(&h3);
    idDelete(&e);
    e=h2;
  }
  idSkipZeroes(e);
  return e;
}

/*2
*the minimal index of used variables - 1
*/
int pLowVar (poly p)
{
  int k,l,lex;

  if (p == NULL) return -1;

  k = 32000;/*a very large dummy value*/
  while (p != NULL)
  {
    l = 1;
    lex = pGetExp(p,l);
    while ((l < pVariables) && (lex == 0))
    {
      l++;
      lex = pGetExp(p,l);
    }
    l--;
    if (l < k) k = l;
    pIter(p);
  }
  return k;
}

/*3
*multiplies p with t (!cas) or  (t-1)
*the index of t is:1, so we have to shift all variables
*p is NOT in the actual ring, it has no t
*/
static poly pMultWithT (poly p,BOOLEAN cas)
{
  /*qp is the working pointer in p*/
  /*result is the result, qresult is the working pointer*/
  /*pp is p in the actual ring(shifted), qpp the working pointer*/
  poly result,qp,pp;
  poly qresult=NULL;
  poly qpp=NULL;
  int  i,j,lex;
  number n;

  pp = NULL;
  result = NULL;
  qp = p;
  while (qp != NULL)
  {
    i = 0;
    if (result == NULL)
    {/*first monomial*/
      result = pInit();
      qresult = result;
    }
    else
    {
      qresult->next = pInit();
      pIter(qresult);
    }
    for (j=pVariables-1; j>0; j--)
    {
      lex = pGetExp(qp,j);
      pSetExp(qresult,j+1,lex);/*copy all variables*/
    }
    lex = pGetComp(qp);
    pSetComp(qresult,lex);
    n=nCopy(pGetCoeff(qp));
    pSetCoeff0(qresult,n);
    qresult->next = NULL;
    pSetm(qresult);
    /*qresult is now qp brought into the actual ring*/
    if (cas)
    { /*case: mult with t-1*/
      pSetExp(qresult,1,0);
      pSetm(qresult);
      if (pp == NULL)
      { /*first monomial*/
        pp = pCopy(qresult);
        qpp = pp;
      }
      else
      {
        qpp->next = pCopy(qresult);
        pIter(qpp);
      }
      pGetCoeff(qpp)=nNeg(pGetCoeff(qpp));
      /*now qpp contains -1*qp*/
    }
    pSetExp(qresult,1,1);/*this is mult. by t*/
    pSetm(qresult);
    pIter(qp);
  }
  /*
  *now p is processed:
  *result contains t*p
  * if cas: pp contains -1*p (in the new ring)
  */
  if (cas)  qresult->next = pp;
  /*  else      qresult->next = NULL;*/
  return result;
}

/*2
* verschiebt die Indizees der Modulerzeugenden um i
*/
void pShift (poly * p,int i)
{
  poly qp1 = *p,qp2 = *p;/*working pointers*/
  int     j = pMaxComp(*p),k = pMinComp(*p);

  if (j+i < 0) return ;
  while (qp1 != NULL)
  {
    if ((pGetComp(qp1)+i > 0) || ((j == -i) && (j == k)))
    {
      pAddComp(qp1,i);
      pSetmComp(qp1);
      qp2 = qp1;
      pIter(qp1);
    }
    else
    {
      if (qp2 == *p)
      {
        pIter(*p);
        pLmDelete(&qp2);
        qp2 = *p;
        qp1 = *p;
      }
      else
      {
        qp2->next = qp1->next;
        if (qp1!=NULL) pLmDelete(&qp1);
        qp1 = qp2->next;
      }
    }
  }
}

/*2
*initialized a field with r numbers between beg and end for the
*procedure idNextChoise
*/
void idInitChoise (int r,int beg,int end,BOOLEAN  *endch,int * choise)
{
  /*returns the first choise of r numbers between beg and end*/
  int i;
  for (i=0; i<r; i++)
  {
    choise[i] = 0;
  }
  if (r <= end-beg+1)
    for (i=0; i<r; i++)
    {
      choise[i] = beg+i;
    }
  if (r > end-beg+1)
    *endch = TRUE;
  else
    *endch = FALSE;
}

/*2
*returns the next choise of r numbers between beg and end
*/
void idGetNextChoise (int r,int end,BOOLEAN  *endch,int * choise)
{
  int i = r-1,j;
  while ((i >= 0) && (choise[i] == end))
  {
    i--;
    end--;
  }
  if (i == -1)
    *endch = TRUE;
  else
  {
    choise[i]++;
    for (j=i+1; j<r; j++)
    {
      choise[j] = choise[i]+j-i;
    }
    *endch = FALSE;
  }
}

/*2
*takes the field choise of d numbers between beg and end, cancels the t-th
*entree and searches for the ordinal number of that d-1 dimensional field
* w.r.t. the algorithm of construction
*/
int idGetNumberOfChoise(int t, int d, int begin, int end, int * choise)
{
  int * localchoise,i,result=0;
  BOOLEAN b=FALSE;

  if (d<=1) return 1;
  localchoise=(int*)omAlloc((d-1)*sizeof(int));
  idInitChoise(d-1,begin,end,&b,localchoise);
  while (!b)
  {
    result++;
    i = 0;
    while ((i<t) && (localchoise[i]==choise[i])) i++;
    if (i>=t)
    {
      i = t+1;
      while ((i<d) && (localchoise[i-1]==choise[i])) i++;
      if (i>=d)
      {
        omFreeSize((ADDRESS)localchoise,(d-1)*sizeof(int));
        return result;
      }
    }
    idGetNextChoise(d-1,end,&b,localchoise);
  }
  omFreeSize((ADDRESS)localchoise,(d-1)*sizeof(int));
  return 0;
}

/*2
*computes the binomial coefficient
*/
int binom (int n,int r)
{
  int i,result;

  if (r==0) return 1;
  if (n-r<r) return binom(n,n-r);
  result = n-r+1;
  for (i=2;i<=r;i++)
  {
    result *= n-r+i;
    if (result<0)
    {
      WarnS("overflow in binomials");
      return 0;
    }
    result /= i;
  }
  return result;
}

/*2
*the free module of rank i
*/
ideal idFreeModule (int i)
{
  int j;
  ideal h;

  h=idInit(i,i);
  for (j=0; j<i; j++)
  {
    h->m[j] = pOne();
    pSetComp(h->m[j],j+1);
    pSetmComp(h->m[j]);
  }
  return h;
}

ideal idSectWithElim (ideal h1,ideal h2)
// does not destroy h1,h2
{
  if (TEST_OPT_PROT) PrintS("intersect by elimination method\n");
  assume(!idIs0(h1));
  assume(!idIs0(h2));
  assume(IDELEMS(h1)<=IDELEMS(h2));
  assume(idRankFreeModule(h1)==0);
  assume(idRankFreeModule(h2)==0);
  // add a new variable:
  int j;
  ring origRing=currRing;
  ring r=rCopy0(origRing);
  r->N++;
  r->block0[0]=1;
  r->block1[0]= r->N;
  omFree(r->order);
  r->order=(int*)omAlloc0(3*sizeof(int*));
  r->order[0]=ringorder_dp;
  r->order[1]=ringorder_C;
  char **names=(char**)omAlloc0(rVar(r) * sizeof(char_ptr));
  for (j=0;j<r->N-1;j++) names[j]=r->names[j];
  names[r->N-1]=omStrDup("@");
  omFree(r->names);
  r->names=names;
  rComplete(r,TRUE);
  // fetch h1, h2
  ideal h;
  h1=idrCopyR(h1,origRing,r);
  h2=idrCopyR(h2,origRing,r);
  // switch to temp. ring r
  rChangeCurrRing(r);
  // create 1-t, t
  poly omt=pOne();
  pSetExp(omt,r->N,1);
  poly t=pCopy(omt);
  pSetm(omt);
  omt=pNeg(omt);
  omt=pAdd(omt,pOne());
  // compute (1-t)*h1
  h1=(ideal)mpMultP((matrix)h1,omt);
  // compute t*h2
  h2=(ideal)mpMultP((matrix)h2,pCopy(t));
  // (1-t)h1 + t*h2
  h=idInit(IDELEMS(h1)+IDELEMS(h2),1);
  int l;
  for (l=IDELEMS(h1)-1; l>=0; l--)
  {
    h->m[l] = h1->m[l];  h1->m[l]=NULL;
  }
  j=IDELEMS(h1);
  for (l=IDELEMS(h2)-1; l>=0; l--)
  {
    h->m[l+j] = h2->m[l];  h2->m[l]=NULL;
  }
  idDelete(&h1);
  idDelete(&h2);
  // eliminate t:

  ideal res=idElimination(h,t);
  // cleanup 
  idDelete(&h);
  res=idrMoveR(res,r,origRing);
  rChangeCurrRing(origRing);
  rKill(r);
  return res;
}
/*2
* h3 := h1 intersect h2
*/
ideal idSect (ideal h1,ideal h2)
{
  int i,j,k,length;
  int flength = idRankFreeModule(h1);
  int slength = idRankFreeModule(h2);
  int rank=si_min(flength,slength);
  if ((idIs0(h1)) || (idIs0(h2)))  return idInit(1,rank);

  ideal first,second,temp,temp1,result;
  poly p,q;

  if (IDELEMS(h1)<IDELEMS(h2))
  {
    first = h1;
    second = h2;
  }
  else
  {
    first = h2;
    second = h1;
    int t=flength; flength=slength; slength=t;
  }
  length  = si_max(flength,slength);
  if (length==0)
  {
    if ((currQuotient==NULL)
    && (currRing->OrdSgn==1)
    && (!rIsPluralRing(currRing))
    && ((TEST_V_INTERSECT_ELIM) || (!TEST_V_INTERSECT_SYZ)))
      return idSectWithElim(first,second);
    else length = 1;
  }
  if (TEST_OPT_PROT) PrintS("intersect by syzygy methods\n");
  j = IDELEMS(first);

  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();
  rSetSyzComp(length);

  while ((j>0) && (first->m[j-1]==NULL)) j--;
  temp = idInit(j /*IDELEMS(first)*/+IDELEMS(second),length+j);
  k = 0;
  for (i=0;i<j;i++)
  {
    if (first->m[i]!=NULL)
    {
      if (syz_ring==orig_ring)
        temp->m[k] = pCopy(first->m[i]);
      else
        temp->m[k] = prCopyR(first->m[i], orig_ring);
      q = pOne();
      pSetComp(q,i+1+length);
      pSetmComp(q);
      if (flength==0) pShift(&(temp->m[k]),1);
      p = temp->m[k];
      while (pNext(p)!=NULL) pIter(p);
      pNext(p) = q;
      k++;
    }
  }
  for (i=0;i<IDELEMS(second);i++)
  {
    if (second->m[i]!=NULL)
    {
      if (syz_ring==orig_ring)
        temp->m[k] = pCopy(second->m[i]);
      else
        temp->m[k] = prCopyR(second->m[i], orig_ring);
      if (slength==0) pShift(&(temp->m[k]),1);
      k++;
    }
  }
  intvec *w=NULL;
  temp1 = kStd(temp,currQuotient,testHomog,&w,NULL,length);
  if (w!=NULL) delete w;
  idDelete(&temp);
  if(syz_ring!=orig_ring)
    rChangeCurrRing(orig_ring);

  result = idInit(IDELEMS(temp1),rank);
  j = 0;
  for (i=0;i<IDELEMS(temp1);i++)
  {
    if ((temp1->m[i]!=NULL)
    && (p_GetComp(temp1->m[i],syz_ring)>length))
    {
      if(syz_ring==orig_ring)
      {
        p = temp1->m[i];
      }
      else
      {
        p = prMoveR(temp1->m[i], syz_ring);
      }
      temp1->m[i]=NULL;
      while (p!=NULL)
      {
        q = pNext(p);
        pNext(p) = NULL;
        k = pGetComp(p)-1-length;
        pSetComp(p,0);
        pSetmComp(p);
        /* Warning! multiply only from the left! it's very important for Plural */
        result->m[j] = pAdd(result->m[j],pMult(p,pCopy(first->m[k])));
        p = q;
      }
      j++;
    }
  }
  if(syz_ring!=orig_ring)
  {
    rChangeCurrRing(syz_ring);
    idDelete(&temp1);
    rChangeCurrRing(orig_ring);
    rKill(syz_ring);
  }
  else
  {
    idDelete(&temp1);
  }

  idSkipZeroes(result);
  if (TEST_OPT_RETURN_SB)
  {
     w=NULL;
     temp1=kStd(result,currQuotient,testHomog,&w);
     if (w!=NULL) delete w;
     idDelete(&result);
     idSkipZeroes(temp1);
     return temp1;
  }
  else //temp1=kInterRed(result,currQuotient);
    return result;
}

/*2
* ideal/module intersection for a list of objects
* given as 'resolvente'
*/
ideal idMultSect(resolvente arg, int length)
{
  int i,j=0,k=0,syzComp,l,maxrk=-1,realrki;
  ideal bigmat,tempstd,result;
  poly p;
  int isIdeal=0;
  intvec * w=NULL;

  /* find 0-ideals and max rank -----------------------------------*/
  for (i=0;i<length;i++)
  {
    if (!idIs0(arg[i]))
    {
      realrki=idRankFreeModule(arg[i]);
      k++;
      j += IDELEMS(arg[i]);
      if (realrki>maxrk) maxrk = realrki;
    }
    else
    {
      if (arg[i]!=NULL)
      {
        return idInit(1,arg[i]->rank);
      }
    }
  }
  if (maxrk == 0)
  {
    isIdeal = 1;
    maxrk = 1;
  }
  /* init -----------------------------------------------------------*/
  j += maxrk;
  syzComp = k*maxrk;

  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();
  rSetSyzComp(syzComp);

  bigmat = idInit(j,(k+1)*maxrk);
  /* create unit matrices ------------------------------------------*/
  for (i=0;i<maxrk;i++)
  {
    for (j=0;j<=k;j++)
    {
      p = pOne();
      pSetComp(p,i+1+j*maxrk);
      pSetmComp(p);
      bigmat->m[i] = pAdd(bigmat->m[i],p);
    }
  }
  /* enter given ideals ------------------------------------------*/
  i = maxrk;
  k = 0;
  for (j=0;j<length;j++)
  {
    if (arg[j]!=NULL)
    {
      for (l=0;l<IDELEMS(arg[j]);l++)
      {
        if (arg[j]->m[l]!=NULL)
        {
          if (syz_ring==orig_ring)
            bigmat->m[i] = pCopy(arg[j]->m[l]);
          else
            bigmat->m[i] = prCopyR(arg[j]->m[l], orig_ring);
          pShift(&(bigmat->m[i]),k*maxrk+isIdeal);
          i++;
        }
      }
      k++;
    }
  }
  /* std computation --------------------------------------------*/
  tempstd = kStd(bigmat,currQuotient,testHomog,&w,NULL,syzComp);
  if (w!=NULL) delete w;
  idDelete(&bigmat);

  if(syz_ring!=orig_ring)
    rChangeCurrRing(orig_ring);

  /* interprete result ----------------------------------------*/
  result = idInit(IDELEMS(tempstd),maxrk);
  k = 0;
  for (j=0;j<IDELEMS(tempstd);j++)
  {
    if ((tempstd->m[j]!=NULL) && (p_GetComp(tempstd->m[j],syz_ring)>syzComp))
    {
      if (syz_ring==orig_ring)
        p = pCopy(tempstd->m[j]);
      else
        p = prCopyR(tempstd->m[j], syz_ring);
      pShift(&p,-syzComp-isIdeal);
      result->m[k] = p;
      k++;
    }
  }
  /* clean up ----------------------------------------------------*/
  if(syz_ring!=orig_ring)
    rChangeCurrRing(syz_ring);
  idDelete(&tempstd);
  if(syz_ring!=orig_ring)
  {
    rChangeCurrRing(orig_ring);
    rKill(syz_ring);
  }
  idSkipZeroes(result);
  return result;
}

/*2
*computes syzygies of h1,
*if quot != NULL it computes in the quotient ring modulo "quot"
*works always in a ring with ringorder_s
*/
static ideal idPrepare (ideal  h1, tHomog hom, int syzcomp, intvec **w)
{
  ideal   h2, h3;
  int     i;
  int     j,jj=0,k;
  poly    p,q;

  if (idIs0(h1)) return NULL;
  k = idRankFreeModule(h1);
  h2=idCopy(h1);
  i = IDELEMS(h2)-1;
  if (k == 0)
  {
    for (j=0; j<=i; j++) pShift(&(h2->m[j]),1);
    k = 1;
  }
  if (syzcomp<k)
  {
    Warn("syzcomp too low, should be %d instead of %d",k,syzcomp);
    syzcomp = k;
    rSetSyzComp(k);
  }
  h2->rank = syzcomp+i+1;

  //if (hom==testHomog)
  //{
  //  if(idHomIdeal(h1,currQuotient))
  //  {
  //    hom=TRUE;
  //  }
  //}

#if MYTEST
#ifdef RDEBUG
  Print("Prepare::h2: ");
  idPrint(h2);

  for(j=0;j<IDELEMS(h2);j++) pTest(h2->m[j]);

#endif
#endif

  for (j=0; j<=i; j++)
  {
    p = h2->m[j];
    q = pOne();
    pSetComp(q,syzcomp+1+j);
    pSetmComp(q);
    if (p!=NULL)
    {
      while (pNext(p)) pIter(p);
      p->next = q;
    }
    else
      h2->m[j]=q;
  }

#ifdef PDEBUG
  for(j=0;j<IDELEMS(h2);j++) pTest(h2->m[j]);

#if MYTEST
#ifdef RDEBUG
  Print("Prepare::Input: ");
  idPrint(h2);

  Print("Prepare::currQuotient: ");
  idPrint(currQuotient);
#endif
#endif

#endif

  idTest(h2);

  h3 = kStd(h2,currQuotient,hom,w,NULL,syzcomp);

#if MYTEST
#ifdef RDEBUG
  Print("Prepare::Output: ");
  idPrint(h3);
  for(j=0;j<IDELEMS(h2);j++) pTest(h3->m[j]);
#endif
#endif


  idDelete(&h2);
  return h3;
}

/*2
* compute the syzygies of h1 in R/quot,
* weights of components are in w
* if setRegularity, return the regularity in deg
* do not change h1,  w
*/
ideal idSyzygies (ideal  h1, tHomog h,intvec **w, BOOLEAN setSyzComp,
                  BOOLEAN setRegularity, int *deg)
{
  ideal s_h1;
  poly  p;
  int   j, k, length=0,reg;
  BOOLEAN isMonomial=TRUE;
  int ii, idElemens_h1;

  assume(h1 != NULL);

  idElemens_h1=IDELEMS(h1);
#ifdef PDEBUG
  for(ii=0;ii<idElemens_h1 /*IDELEMS(h1)*/;ii++) pTest(h1->m[ii]);
#endif
  if (idIs0(h1))
  {
    ideal result=idFreeModule(idElemens_h1/*IDELEMS(h1)*/);
    int curr_syz_limit=rGetCurrSyzLimit();
    if (curr_syz_limit>0)
    for (ii=0;ii<idElemens_h1/*IDELEMS(h1)*/;ii++)
    {
      if (h1->m[ii]!=NULL)
        pShift(&h1->m[ii],curr_syz_limit);
    }
    return result;
  }
  int slength=(int)idRankFreeModule(h1);
  k=si_max(1,slength /*idRankFreeModule(h1)*/);

  assume(currRing != NULL);
  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();

  if (setSyzComp)
    rSetSyzComp(k);

  if (orig_ring != syz_ring)
  {
    s_h1=idrCopyR_NoSort(h1,orig_ring);
  }
  else
  {
    s_h1 = h1;
  }

  idTest(s_h1);

  ideal s_h3=idPrepare(s_h1,h,k,w); // main (syz) GB computation

  if (s_h3==NULL)
  {
    return idFreeModule( idElemens_h1 /*IDELEMS(h1)*/);
  }

  if (orig_ring != syz_ring)
  {
    idDelete(&s_h1);
    for (j=0; j<IDELEMS(s_h3); j++)
    {
      if (s_h3->m[j] != NULL)
      {
        if (p_MinComp(s_h3->m[j],syz_ring) > k)
          pShift(&s_h3->m[j], -k);
        else
          pDelete(&s_h3->m[j]);
      }
    }
    idSkipZeroes(s_h3);
    s_h3->rank -= k;
    rChangeCurrRing(orig_ring);
    s_h3 = idrMoveR_NoSort(s_h3, syz_ring);
    rKill(syz_ring);
    #ifdef HAVE_PLURAL
    if (rIsPluralRing(currRing))
    {
      idDelMultiples(s_h3);
      idSkipZeroes(s_h3);
    }
    #endif
    idTest(s_h3);
    return s_h3;
  }

  ideal e = idInit(IDELEMS(s_h3), s_h3->rank);

  for (j=IDELEMS(s_h3)-1; j>=0; j--)
  {
    if (s_h3->m[j] != NULL)
    {
      if (p_MinComp(s_h3->m[j],syz_ring) <= k)
      {
        e->m[j] = s_h3->m[j];
        isMonomial=isMonomial && (pNext(s_h3->m[j])==NULL);
        pDelete(&pNext(s_h3->m[j]));
        s_h3->m[j] = NULL;
      }
    }
  }

  idSkipZeroes(s_h3);
  idSkipZeroes(e);

  if ((deg != NULL)
  && (!isMonomial)
  && (!TEST_OPT_NOTREGULARITY)
  && (setRegularity)
  && (h==isHomog)
  && (!rIsPluralRing(currRing))
  )
  {
    ring dp_C_ring = rCurrRingAssure_dp_C();
    if (dp_C_ring != syz_ring)
      e = idrMoveR_NoSort(e, syz_ring);
    resolvente res = sySchreyerResolvente(e,-1,&length,TRUE, TRUE);
    intvec * dummy = syBetti(res,length,&reg, *w);
    *deg = reg+2;
    delete dummy;
    for (j=0;j<length;j++)
    {
      if (res[j]!=NULL) idDelete(&(res[j]));
    }
    omFreeSize((ADDRESS)res,length*sizeof(ideal));
    idDelete(&e);
    if (dp_C_ring != syz_ring)
    {
      rChangeCurrRing(syz_ring);
      rKill(dp_C_ring);
    }
  }
  else
  {
    idDelete(&e);
  }
  idTest(s_h3);
  if (currQuotient != NULL)
  {
    ideal ts_h3=kStd(s_h3,currQuotient,h,w);
    idDelete(&s_h3);
    s_h3 = ts_h3;
  }
  return s_h3;
}

/*2
*/
ideal idXXX (ideal  h1, int k)
{
  ideal s_h1;
  int j;
  intvec *w=NULL;

  assume(currRing != NULL);
  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();

  rSetSyzComp(k);

  if (orig_ring != syz_ring)
  {
    s_h1=idrCopyR_NoSort(h1,orig_ring);
  }
  else
  {
    s_h1 = h1;
  }

  ideal s_h3=kStd(s_h1,NULL,testHomog,&w,NULL,k);

  if (s_h3==NULL)
  {
    return idFreeModule(IDELEMS(h1));
  }

  if (orig_ring != syz_ring)
  {
    idDelete(&s_h1);
    idSkipZeroes(s_h3);
    rChangeCurrRing(orig_ring);
    s_h3 = idrMoveR_NoSort(s_h3, syz_ring);
    rKill(syz_ring);
    idTest(s_h3);
    return s_h3;
  }

  idSkipZeroes(s_h3);
  idTest(s_h3);
  return s_h3;
}

/*
*computes a standard basis for h1 and stores the transformation matrix
* in ma
*/
ideal idLiftStd (ideal  h1, matrix* ma, tHomog hi, ideal * syz)
{
  int   i, j, k, t, inputIsIdeal=idRankFreeModule(h1);
  poly  p=NULL, q, qq;
  intvec *w=NULL;

  idDelete((ideal*)ma);
  BOOLEAN lift3=FALSE;
  if (syz!=NULL) { lift3=TRUE; idDelete(syz); }
  if (idIs0(h1))
  {
    *ma=mpNew(1,0);
    if (lift3)
    {
      *syz=idFreeModule(IDELEMS(h1));
      int curr_syz_limit=rGetCurrSyzLimit();
      if (curr_syz_limit>0)
      for (int ii=0;ii<IDELEMS(h1);ii++)
      {
        if (h1->m[ii]!=NULL)
          pShift(&h1->m[ii],curr_syz_limit);
      }
    }
    return idInit(1,h1->rank);
  }

  BITSET save_verbose=verbose;

  k=si_max(1,(int)idRankFreeModule(h1));

  if ((k==1) && (!lift3)) verbose |=Sy_bit(V_IDLIFT);

  ring orig_ring = currRing;
  ring syz_ring = rCurrRingAssure_SyzComp();
  rSetSyzComp(k);

  ideal s_h1=h1;

  if (orig_ring != syz_ring)
    s_h1 = idrCopyR_NoSort(h1,orig_ring);
  else
    s_h1 = h1;

  ideal s_h3=idPrepare(s_h1,hi,k,&w); // main (syz) GB computation

  ideal s_h2 = idInit(IDELEMS(s_h3), s_h3->rank);

  if (lift3) (*syz)=idInit(IDELEMS(s_h3),IDELEMS(h1));

  if (w!=NULL) delete w;
  i = 0;

  // now sort the result, SB : leave in s_h3
  //                      T:  put in s_h2
  //                      syz: put in *syz
  for (j=0; j<IDELEMS(s_h3); j++)
  {
    if (s_h3->m[j] != NULL)
    {
      //if (p_MinComp(s_h3->m[j],syz_ring) <= k)
      if (pGetComp(s_h3->m[j]) <= k) // syz_ring == currRing
      {
        i++;
        q = s_h3->m[j];
        while (pNext(q) != NULL)
        {
          if (pGetComp(pNext(q)) > k)
          {
            s_h2->m[j] = pNext(q);
            pNext(q) = NULL;
          }
          else
          {
            pIter(q);
          }
        }
        if (!inputIsIdeal) pShift(&(s_h3->m[j]), -1);
      }
      else
      {
        // we a syzygy here:
        if (lift3)
        {
          pShift(&s_h3->m[j], -k);
          (*syz)->m[j]=s_h3->m[j];
          s_h3->m[j]=NULL;
        }
        else
          pDelete(&(s_h3->m[j]));
      }
    }
  }
  idSkipZeroes(s_h3);
  //extern char * iiStringMatrix(matrix im, int dim,char ch);
  //PrintS("SB: ----------------------------------------\n");
  //PrintS(iiStringMatrix((matrix)s_h3,k,'\n'));
  //PrintLn();
  //PrintS("T: ----------------------------------------\n");
  //PrintS(iiStringMatrix((matrix)s_h2,h1->rank,'\n'));
  //PrintLn();

  if (lift3) idSkipZeroes(*syz);

  j = IDELEMS(s_h1);


  if (syz_ring!=orig_ring)
  {
    idDelete(&s_h1);
    rChangeCurrRing(orig_ring);
  }

  *ma = mpNew(j,i);

  i = 1;
  for (j=0; j<IDELEMS(s_h2); j++)
  {
    if (s_h2->m[j] != NULL)
    {
      q = prMoveR( s_h2->m[j], syz_ring);
      s_h2->m[j] = NULL;

      while (q != NULL)
      {
        p = q;
        pIter(q);
        pNext(p) = NULL;
        t=pGetComp(p);
        pSetComp(p,0);
        pSetmComp(p);
        MATELEM(*ma,t-k,i) = pAdd(MATELEM(*ma,t-k,i),p);
      }
      i++;
    }
  }
  idDelete(&s_h2);

  for (i=0; i<IDELEMS(s_h3); i++)
  {
    s_h3->m[i] = prMoveR_NoSort(s_h3->m[i], syz_ring);
  }
  if (lift3)
  {
    for (i=0; i<IDELEMS(*syz); i++)
    {
      (*syz)->m[i] = prMoveR_NoSort((*syz)->m[i], syz_ring);
    }
  }

  if (syz_ring!=orig_ring) rKill(syz_ring);
  verbose = save_verbose;
  return s_h3;
}

static void idPrepareStd(ideal s_temp, int k)
{
  int j,rk=idRankFreeModule(s_temp);
  poly p,q;

  if (rk == 0)
  {
    for (j=0; j<IDELEMS(s_temp); j++)
    {
      if (s_temp->m[j]!=NULL) pSetCompP(s_temp->m[j],1);
    }
    k = si_max(k,1);
  }
  for (j=0; j<IDELEMS(s_temp); j++)
  {
    if (s_temp->m[j]!=NULL)
    {
      p = s_temp->m[j];
      q = pOne();
      //pGetCoeff(q)=nNeg(pGetCoeff(q));   //set q to -1
      pSetComp(q,k+1+j);
      pSetmComp(q);
      while (pNext(p)) pIter(p);
      pNext(p) = q;
    }
  }
}

/*2
*computes a representation of the generators of submod with respect to those
* of mod
*/

ideal idLift(ideal mod, ideal submod,ideal *rest, BOOLEAN goodShape,
             BOOLEAN isSB, BOOLEAN divide, matrix *unit)
{
  int lsmod =idRankFreeModule(submod), i, j, k;
  int comps_to_add=0;
  poly p;

  if (idIs0(submod))
  {
    if (unit!=NULL)
    {
      *unit=mpNew(1,1);
      MATELEM(*unit,1,1)=pOne();
    }
    if (rest!=NULL)
    {
      *rest=idInit(1,mod->rank);
    }
    return idInit(1,mod->rank);
  }
  if (idIs0(mod)) /* and not idIs0(submod) */
  {
    WerrorS("2nd module does not lie in the first");
    #if 0
    if (unit!=NULL)
    {
      i=IDELEMS(submod);
      *unit=mpNew(i,i);
      for (j=i;j>0;j--)
      {
        MATELEM(*unit,j,j)=pOne();
      }
    }
    if (rest!=NULL)
    {
      *rest=idCopy(submod);
    }
    return idInit(1,mod->rank);
    #endif
    return idInit(IDELEMS(submod),submod->rank);
  }
  if (unit!=NULL)
  {
    comps_to_add = IDELEMS(submod);
    while ((comps_to_add>0) && (submod->m[comps_to_add-1]==NULL))
      comps_to_add--;
  }
  k=si_max(idRankFreeModule(mod),idRankFreeModule(submod));
  if  ((k!=0) && (lsmod==0)) lsmod=1;
  k=si_max(k,(int)mod->rank);
  if (k<submod->rank) { WarnS("rk(submod) > rk(mod) ?");k=submod->rank; }

  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();
  rSetSyzComp(k);

  ideal s_mod, s_temp;
  if (orig_ring != syz_ring)
  {
    s_mod = idrCopyR_NoSort(mod,orig_ring);
    s_temp = idrCopyR_NoSort(submod,orig_ring);
  }
  else
  {
    s_mod = mod;
    s_temp = idCopy(submod);
  }
  ideal s_h3;
  if (isSB)
  {
    s_h3 = idCopy(s_mod);
    idPrepareStd(s_h3, k+comps_to_add);
  }
  else
  {
    s_h3 = idPrepare(s_mod,(tHomog)FALSE,k+comps_to_add,NULL);
  }
  if (!goodShape)
  {
    for (j=0;j<IDELEMS(s_h3);j++)
    {
      if ((s_h3->m[j] != NULL) && (pMinComp(s_h3->m[j]) > k))
        pDelete(&(s_h3->m[j]));
    }
  }
  idSkipZeroes(s_h3);
  if (lsmod==0)
  {
    for (j=IDELEMS(s_temp);j>0;j--)
    {
      if (s_temp->m[j-1]!=NULL)
        pShift(&(s_temp->m[j-1]),1);
    }
  }
  if (unit!=NULL)
  {
    for(j = 0;j<comps_to_add;j++)
    {
      p = s_temp->m[j];
      if (p!=NULL)
      {
        while (pNext(p)!=NULL) pIter(p);
        pNext(p) = pOne();
        pIter(p);
        pSetComp(p,1+j+k);
        pSetmComp(p);
        p = pNeg(p);
      }
    }
  }
  ideal s_result = kNF(s_h3,currQuotient,s_temp,k);
  s_result->rank = s_h3->rank;
  ideal s_rest = idInit(IDELEMS(s_result),k);
  idDelete(&s_h3);
  idDelete(&s_temp);

  for (j=0;j<IDELEMS(s_result);j++)
  {
    if (s_result->m[j]!=NULL)
    {
      if (pGetComp(s_result->m[j])<=k)
      {
        if (!divide)
        {
          if (isSB)
          {
            WarnS("first module not a standardbasis\n"
              "// ** or second not a proper submodule");
          }
          else
            WerrorS("2nd module does not lie in the first");
          idDelete(&s_result);
          idDelete(&s_rest);
          s_result=idInit(IDELEMS(submod),submod->rank);
          break;
        }
        else
        {
          p = s_rest->m[j] = s_result->m[j];
          while ((pNext(p)!=NULL) && (pGetComp(pNext(p))<=k)) pIter(p);
          s_result->m[j] = pNext(p);
          pNext(p) = NULL;
        }
      }
      pShift(&(s_result->m[j]),-k);
      pNeg(s_result->m[j]);
    }
  }
  if ((lsmod==0) && (!idIs0(s_rest)))
  {
    for (j=IDELEMS(s_rest);j>0;j--)
    {
      if (s_rest->m[j-1]!=NULL)
      {
        pShift(&(s_rest->m[j-1]),-1);
        s_rest->m[j-1] = s_rest->m[j-1];
      }
    }
  }
  if(syz_ring!=orig_ring)
  {
    idDelete(&s_mod);
    rChangeCurrRing(orig_ring);
    s_result = idrMoveR_NoSort(s_result, syz_ring);
    s_rest = idrMoveR_NoSort(s_rest, syz_ring);
    rKill(syz_ring);
  }
  if (rest!=NULL)
    *rest = s_rest;
  else
    idDelete(&s_rest);
//idPrint(s_result);
  if (unit!=NULL)
  {
    *unit=mpNew(comps_to_add,comps_to_add);
    int i;
    for(i=0;i<IDELEMS(s_result);i++)
    {
      poly p=s_result->m[i];
      poly q=NULL;
      while(p!=NULL)
      {
        if(pGetComp(p)<=comps_to_add)
        {
          pSetComp(p,0);
          if (q!=NULL)
          {
            pNext(q)=pNext(p);
          }
          else
          {
            pIter(s_result->m[i]);
          }
          pNext(p)=NULL;
          MATELEM(*unit,i+1,i+1)=pAdd(MATELEM(*unit,i+1,i+1),p);
          if(q!=NULL)   p=pNext(q);
          else          p=s_result->m[i];
        }
        else
        {
          q=p;
          pIter(p);
        }
      }
      pShift(&s_result->m[i],-comps_to_add);
    }
  }
  return s_result;
}

/*2
*computes division of P by Q with remainder up to (w-weighted) degree n
*P, Q, and w are not changed
*/
void idLiftW(ideal P,ideal Q,int n,matrix &T, ideal &R,short *w)
{
  long N=0;
  int i;
  for(i=IDELEMS(Q)-1;i>=0;i--)
    if(w==NULL)
      N=si_max(N,pDeg(Q->m[i]));
    else
      N=si_max(N,pDegW(Q->m[i],w));
  N+=n;

  T=mpNew(IDELEMS(Q),IDELEMS(P));
  R=idInit(IDELEMS(P),P->rank);

  for(i=IDELEMS(P)-1;i>=0;i--)
  {
    poly p;
    if(w==NULL)
      p=ppJet(P->m[i],N);
    else
      p=ppJetW(P->m[i],N,w);

    int j=IDELEMS(Q)-1;
    while(p!=NULL)
    {
      if(pDivisibleBy(Q->m[j],p))
      {
        poly p0=pDivideM(pHead(p),pHead(Q->m[j]));
        if(w==NULL)
          p=pJet(pSub(p,ppMult_mm(Q->m[j],p0)),N);
        else
          p=pJetW(pSub(p,ppMult_mm(Q->m[j],p0)),N,w);
        pNormalize(p);
        if((w==NULL)&&(pDeg(p0)>n)||(w!=NULL)&&(pDegW(p0,w)>n))
          pDelete(&p0);
        else
          MATELEM(T,j+1,i+1)=pAdd(MATELEM(T,j+1,i+1),p0);
        j=IDELEMS(Q)-1;
      }
      else
      {
        if(j==0)
        {
          poly p0=p;
          pIter(p);
          pNext(p0)=NULL;
          if(((w==NULL)&&(pDeg(p0)>n))
          ||((w!=NULL)&&(pDegW(p0,w)>n)))
            pDelete(&p0);
          else
            R->m[i]=pAdd(R->m[i],p0);
          j=IDELEMS(Q)-1;
        }
        else
          j--;
      }
    }
  }
}

/*2
*computes the quotient of h1,h2 : internal routine for idQuot
*BEWARE: the returned ideals may contain incorrectly ordered polys !
*
*/
static ideal idInitializeQuot (ideal  h1, ideal h2, BOOLEAN h1IsStb,
                               BOOLEAN *addOnlyOne, int *kkmax)
{
  ideal temph1;
  poly     p,q = NULL;
  int i,l,ll,k,kkk,kmax;
  int j = 0;
  int k1 = idRankFreeModule(h1);
  int k2 = idRankFreeModule(h2);
  tHomog   hom=isNotHomog;

  k=si_max(k1,k2);
  if (k==0)
    k = 1;
  if ((k2==0) && (k>1)) *addOnlyOne = FALSE;

  intvec * weights;
  hom = (tHomog)idHomModule(h1,currQuotient,&weights);
  if (/**addOnlyOne &&*/ (!h1IsStb))
    temph1 = kStd(h1,currQuotient,hom,&weights,NULL);
  else
    temph1 = idCopy(h1);
  if (weights!=NULL) delete weights;
  idTest(temph1);
/*--- making a single vector from h2 ---------------------*/
  for (i=0; i<IDELEMS(h2); i++)
  {
    if (h2->m[i] != NULL)
    {
      p = pCopy(h2->m[i]);
      if (k2 == 0)
        pShift(&p,j*k+1);
      else
        pShift(&p,j*k);
      q = pAdd(q,p);
      j++;
    }
  }
  *kkmax = kmax = j*k+1;
/*--- adding a monomial for the result (syzygy) ----------*/
  p = q;
  while (pNext(p)!=NULL) pIter(p);
  pNext(p) = pOne();
  pIter(p);
  pSetComp(p,kmax);
  pSetmComp(p);
/*--- constructing the big matrix ------------------------*/
  ideal h4 = idInit(16,kmax+k-1);
  h4->m[0] = q;
  if (k2 == 0)
  {
    if (k > IDELEMS(h4))
    {
      pEnlargeSet(&(h4->m),IDELEMS(h4),k-IDELEMS(h4));
      IDELEMS(h4) = k;
    }
    for (i=1; i<k; i++)
    {
      if (h4->m[i-1]!=NULL)
      {
        p = pCopy_noCheck(h4->m[i-1]);
        pShift(&p,1);
        h4->m[i] = p;
      }
    }
  }
  idSkipZeroes(h4);
  kkk = IDELEMS(h4);
  i = IDELEMS(temph1);
  for (l=0; l<i; l++)
  {
    if(temph1->m[l]!=NULL)
    {
      for (ll=0; ll<j; ll++)
      {
        p = pCopy(temph1->m[l]);
        if (k1 == 0)
          pShift(&p,ll*k+1);
        else
          pShift(&p,ll*k);
        if (kkk >= IDELEMS(h4))
        {
          pEnlargeSet(&(h4->m),IDELEMS(h4),16);
          IDELEMS(h4) += 16;
        }
        h4->m[kkk] = p;
        kkk++;
      }
    }
  }
/*--- if h2 goes in as single vector - the h1-part is just SB ---*/
  if (*addOnlyOne)
  {
    idSkipZeroes(h4);
    p = h4->m[0];
    for (i=0;i<IDELEMS(h4)-1;i++)
    {
      h4->m[i] = h4->m[i+1];
    }
    h4->m[IDELEMS(h4)-1] = p;
    test |= Sy_bit(OPT_SB_1);
  }
  idDelete(&temph1);
  return h4;
}
/*2
*computes the quotient of h1,h2
*/
ideal idQuot (ideal  h1, ideal h2, BOOLEAN h1IsStb, BOOLEAN resultIsIdeal)
{
  // first check for special case h1:(0)
  if (idIs0(h2))
  {
    ideal res;
    if (resultIsIdeal)
    {
      res = idInit(1,1);
      res->m[0] = pOne();
    }
    else
      res = idFreeModule(h1->rank);
    return res;
  }
  BITSET old_test=test;
  int i,l,ll,k,kkk,kmax;
  BOOLEAN  addOnlyOne=TRUE;
  tHomog   hom=isNotHomog;
  intvec * weights1;

  ideal s_h4 = idInitializeQuot (h1,h2,h1IsStb,&addOnlyOne,&kmax);

  hom = (tHomog)idHomModule(s_h4,currQuotient,&weights1);

  ring orig_ring=currRing;
  ring syz_ring=rCurrRingAssure_SyzComp();
  rSetSyzComp(kmax-1);
  if (orig_ring!=syz_ring)
  //  s_h4 = idrMoveR_NoSort(s_h4,orig_ring);
    s_h4 = idrMoveR(s_h4,orig_ring);
  idTest(s_h4);
  #if 0
  void ipPrint_MA0(matrix m, const char *name);
  matrix m=idModule2Matrix(idCopy(s_h4));
  PrintS("start:\n");
  ipPrint_MA0(m,"Q");
  idDelete((ideal *)&m);
  PrintS("last elem:");wrp(s_h4->m[IDELEMS(s_h4)-1]);PrintLn();
  #endif
  ideal s_h3;
  if (addOnlyOne)
  {
    s_h3 = kStd(s_h4,currQuotient,hom,&weights1,NULL,0/*kmax-1*/,IDELEMS(s_h4)-1);
  }
  else
  {
    s_h3 = kStd(s_h4,currQuotient,hom,&weights1,NULL,kmax-1);
  }
  test = old_test;
  #if 0
  // only together with the above debug stuff
  idSkipZeroes(s_h3);
  m=idModule2Matrix(idCopy(s_h3));
  Print("result, kmax=%d:\n",kmax);
  ipPrint_MA0(m,"S");
  idDelete((ideal *)&m);
  #endif
  idTest(s_h3);
  if (weights1!=NULL) delete weights1;
  idDelete(&s_h4);

  for (i=0;i<IDELEMS(s_h3);i++)
  {
    if ((s_h3->m[i]!=NULL) && (pGetComp(s_h3->m[i])>=kmax))
    {
      if (resultIsIdeal)
        pShift(&s_h3->m[i],-kmax);
      else
        pShift(&s_h3->m[i],-kmax+1);
    }
    else
      pDelete(&s_h3->m[i]);
  }
  if (resultIsIdeal)
    s_h3->rank = 1;
  else
    s_h3->rank = h1->rank;
  if(syz_ring!=orig_ring)
  {
    rChangeCurrRing(orig_ring);
    s_h3 = idrMoveR_NoSort(s_h3, syz_ring);
    rKill(syz_ring);
  }
  idSkipZeroes(s_h3);
  idTest(s_h3);
  return s_h3;
}

/*2
*computes recursively all monomials of a certain degree
*in every step the actvar-th entry in the exponential
*vector is incremented and the other variables are
*computed by recursive calls of makemonoms
*if the last variable is reached, the difference to the
*degree is computed directly
*vars is the number variables
*actvar is the actual variable to handle
*deg is the degree of the monomials to compute
*monomdeg is the actual degree of the monomial in consideration
*/
static void makemonoms(int vars,int actvar,int deg,int monomdeg)
{
  poly p;
  int i=0;

  if ((idpowerpoint == 0) && (actvar ==1))
  {
    idpower[idpowerpoint] = pOne();
    monomdeg = 0;
  }
  while (i<=deg)
  {
    if (deg == monomdeg)
    {
      pSetm(idpower[idpowerpoint]);
  …