source: git/kernel/GBEngine/kspoly.cc @ 70a107

/****************************************
*  Computer Algebra System SINGULAR     *
****************************************/
/*
*  ABSTRACT -  Routines for Spoly creation and reductions
*/

// #define PDEBUG 2



#include "kernel/mod2.h"
#include "misc/options.h"
#include "kernel/GBEngine/kutil.h"
#include "coeffs/numbers.h"
#include "polys/monomials/p_polys.h"
#include "polys/templates/p_Procs.h"
#include "polys/nc/nc.h"
#ifdef HAVE_RINGS
#include "kernel/polys.h"
#endif

#ifdef KDEBUG
int red_count = 0;
int create_count = 0;
// define this if reductions are reported on TEST_OPT_DEBUG
#define TEST_OPT_DEBUG_RED
#endif

/***************************************************************
 *
 * Reduces PR with PW
 * Assumes PR != NULL, PW != NULL, Lm(PW) divides Lm(PR)
 *
 * returns 0: okay
 *         1: tailRing changed
 *         -1: cannot change tailRing
 *         2: cannot change tailRing: strat==NULL
 *
 ***************************************************************/
int ksReducePoly(LObject* PR,
                 TObject* PW,
                 poly spNoether,
                 number *coef,
                 kStrategy strat)
{
#ifdef KDEBUG
  red_count++;
#ifdef TEST_OPT_DEBUG_RED
//  if (TEST_OPT_DEBUG)
//  {
//    Print("Red %d:", red_count); PR->wrp(); Print(" with:");
//    PW->wrp();
//    //printf("\necart(PR)-ecart(PW): %i\n",PR->ecart-PW->ecart);
//    //pWrite(PR->p);
//  }
#endif
#endif
  int ret = 0;
  ring tailRing = PR->tailRing;
  kTest_L(PR);
  kTest_T(PW);

  poly p1 = PR->GetLmTailRing();   // p2 | p1
  poly p2 = PW->GetLmTailRing();   // i.e. will reduce p1 with p2; lm = LT(p1) / LM(p2)
  poly t2 = pNext(p2), lm = p1;    // t2 = p2 - LT(p2); really compute P = LC(p2)*p1 - LT(p1)/LM(p2)*p2
  assume(p1 != NULL && p2 != NULL);// Attention, we have rings and there LC(p2) and LC(p1) are special
  p_CheckPolyRing(p1, tailRing);
  p_CheckPolyRing(p2, tailRing);

  pAssume1(p2 != NULL && p1 != NULL &&
           p_DivisibleBy(p2,  p1, tailRing));

  pAssume1(p_GetComp(p1, tailRing) == p_GetComp(p2, tailRing) ||
           (p_GetComp(p2, tailRing) == 0 &&
            p_MaxComp(pNext(p2),tailRing) == 0));

#ifdef HAVE_PLURAL
  if (rIsPluralRing(currRing))
  {
    // for the time being: we know currRing==strat->tailRing
    // no exp-bound checking needed
    // (only needed if exp-bound(tailring)<exp-b(currRing))
    if (PR->bucket!=NULL)  nc_kBucketPolyRed_Z(PR->bucket, p2,coef);
    else
    {
      poly _p = (PR->t_p != NULL ? PR->t_p : PR->p);
      assume(_p != NULL);
      nc_PolyPolyRed(_p, p2,coef, currRing);
      if (PR->t_p!=NULL) PR->t_p=_p; else PR->p=_p;
      PR->pLength=0; // usually not used, GetpLength re-computes it if needed
    }
    return 0;
  }
#endif

  if (t2==NULL)           // Divisor is just one term, therefore it will
  {                       // just cancel the leading term
    PR->LmDeleteAndIter();
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
    return 0;
  }

  p_ExpVectorSub(lm, p2, tailRing); // Calculate the Monomial we must multiply to p2

  //if (tailRing != currRing)
  {
    // check that reduction does not violate exp bound
    while (PW->max_exp != NULL && !p_LmExpVectorAddIsOk(lm, PW->max_exp, tailRing))
    {
      // undo changes of lm
      p_ExpVectorAdd(lm, p2, tailRing);
      if (strat == NULL) return 2;
      if (! kStratChangeTailRing(strat, PR, PW)) return -1;
      tailRing = strat->tailRing;
      p1 = PR->GetLmTailRing();
      p2 = PW->GetLmTailRing();
      t2 = pNext(p2);
      lm = p1;
      p_ExpVectorSub(lm, p2, tailRing);
      ret = 1;
    }
  }

  // take care of coef buisness
  if (! n_IsOne(pGetCoeff(p2), tailRing->cf))
  {
    number bn = pGetCoeff(lm);
    number an = pGetCoeff(p2);
    int ct = ksCheckCoeff(&an, &bn, tailRing->cf);    // Calculate special LC
    p_SetCoeff(lm, bn, tailRing);
    if ((ct == 0) || (ct == 2))
      PR->Tail_Mult_nn(an);
    if (coef != NULL) *coef = an;
    else n_Delete(&an, tailRing->cf);
  }
  else
  {
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
  }


  // and finally,
  PR->Tail_Minus_mm_Mult_qq(lm, t2, pLength(t2) /*PW->GetpLength() - 1*/, spNoether);
  assume(PW->GetpLength() == pLength(PW->p != NULL ? PW->p : PW->t_p));
  PR->LmDeleteAndIter();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if ( (currRing->isLPring) && (!strat->homog) )
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif

#if defined(KDEBUG) && defined(TEST_OPT_DEBUG_RED)
  if (TEST_OPT_DEBUG)
  {
    Print(" to: "); PR->wrp(); Print("\n");
    //printf("\nt^%i ", PR->ecart);pWrite(pHead(PR->p));
  }
#endif
  return ret;
}

int ksReducePolyBound(LObject* PR,
                 TObject* PW,
                 int bound,
                 poly spNoether,
                 number *coef,
                 kStrategy strat)
{
#ifdef KDEBUG
  red_count++;
#ifdef TEST_OPT_DEBUG_RED
  if (TEST_OPT_DEBUG)
  {
    Print("Red %d:", red_count); PR->wrp(); Print(" with:");
    PW->wrp();
    //printf("\necart(PR)-ecart(PW): %i\n",PR->ecart-PW->ecart);
    //pWrite(PR->p);
  }
#endif
#endif
  int ret = 0;
  ring tailRing = PR->tailRing;
  kTest_L(PR);
  kTest_T(PW);

  poly p1 = PR->GetLmTailRing();   // p2 | p1
  poly p2 = PW->GetLmTailRing();   // i.e. will reduce p1 with p2; lm = LT(p1) / LM(p2)
  poly t2 = pNext(p2), lm = p1;    // t2 = p2 - LT(p2); really compute P = LC(p2)*p1 - LT(p1)/LM(p2)*p2
  assume(p1 != NULL && p2 != NULL);// Attention, we have rings and there LC(p2) and LC(p1) are special
  p_CheckPolyRing(p1, tailRing);
  p_CheckPolyRing(p2, tailRing);

  pAssume1(p2 != NULL && p1 != NULL &&
           p_DivisibleBy(p2,  p1, tailRing));

  pAssume1(p_GetComp(p1, tailRing) == p_GetComp(p2, tailRing) ||
           (p_GetComp(p2, tailRing) == 0 &&
            p_MaxComp(pNext(p2),tailRing) == 0));

#ifdef HAVE_PLURAL
  if (rIsPluralRing(currRing))
  {
    // for the time being: we know currRing==strat->tailRing
    // no exp-bound checking needed
    // (only needed if exp-bound(tailring)<exp-b(currRing))
    if (PR->bucket!=NULL)  nc_kBucketPolyRed_Z(PR->bucket, p2,coef);
    else
    {
      poly _p = (PR->t_p != NULL ? PR->t_p : PR->p);
      assume(_p != NULL);
      nc_PolyPolyRed(_p, p2,coef, currRing);
      if (PR->t_p!=NULL) PR->t_p=_p; else PR->p=_p;
      PR->pLength=0; // usually not used, GetpLength re-computes it if needed
    }
    return 0;
  }
#endif

  if (t2==NULL)           // Divisor is just one term, therefore it will
  {                       // just cancel the leading term
    PR->LmDeleteAndIter();
    if (coef != NULL) *coef = n_Init(1, tailRing);
    return 0;
  }

  p_ExpVectorSub(lm, p2, tailRing); // Calculate the Monomial we must multiply to p2

  if (tailRing != currRing)
  {
    // check that reduction does not violate exp bound
    while (PW->max_exp != NULL && !p_LmExpVectorAddIsOk(lm, PW->max_exp, tailRing))
    {
      // undo changes of lm
      p_ExpVectorAdd(lm, p2, tailRing);
      if (strat == NULL) return 2;
      if (! kStratChangeTailRing(strat, PR, PW)) return -1;
      tailRing = strat->tailRing;
      p1 = PR->GetLmTailRing();
      p2 = PW->GetLmTailRing();
      t2 = pNext(p2);
      lm = p1;
      p_ExpVectorSub(lm, p2, tailRing);
      ret = 1;
    }
  }

  // take care of coef buisness
  if (! n_IsOne(pGetCoeff(p2), tailRing))
  {
    number bn = pGetCoeff(lm);
    number an = pGetCoeff(p2);
    int ct = ksCheckCoeff(&an, &bn, tailRing->cf);    // Calculate special LC
    p_SetCoeff(lm, bn, tailRing);
    if ((ct == 0) || (ct == 2))
      PR->Tail_Mult_nn(an);
    if (coef != NULL) *coef = an;
    else n_Delete(&an, tailRing);
  }
  else
  {
    if (coef != NULL) *coef = n_Init(1, tailRing);
  }


  // and finally,
  PR->Tail_Minus_mm_Mult_qq(lm, t2, pLength(t2) /*PW->GetpLength() - 1*/, spNoether);
  assume(PW->GetpLength() == pLength(PW->p != NULL ? PW->p : PW->t_p));
  PR->LmDeleteAndIter();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if ( (currRing->isLPring) && (!strat->homog) )
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif

#if defined(KDEBUG) && defined(TEST_OPT_DEBUG_RED)
  if (TEST_OPT_DEBUG)
  {
    Print(" to: "); PR->wrp(); Print("\n");
    //printf("\nt^%i ", PR->ecart);pWrite(pHead(PR->p));
  }
#endif
  return ret;
}

/***************************************************************
 *
 * Reduces PR with PW
 * Assumes PR != NULL, PW != NULL, Lm(PW) divides Lm(PR)
 *
 ***************************************************************/

int ksReducePolySig(LObject* PR,
                 TObject* PW,
                 long /*idx*/,
                 poly spNoether,
                 number *coef,
                 kStrategy strat)
{
#ifdef KDEBUG
  red_count++;
#ifdef TEST_OPT_DEBUG_RED
  if (TEST_OPT_DEBUG)
  {
    Print("Red %d:", red_count); PR->wrp(); Print(" with:");
    PW->wrp();
  }
#endif
#endif
  int ret = 0;
  ring tailRing = PR->tailRing;
  kTest_L(PR);
  kTest_T(PW);

  // signature-based stuff:
  // checking for sig-safeness first
  // NOTE: This has to be done in the current ring
  //
  /**********************************************
   *
   * TODO:
   * --------------------------------------------
   * if strat->sbaOrder == 1
   * Since we are subdividing lower index and
   * current index reductions it is enough to
   * look at the polynomial part of the signature
   * for a check. This should speed-up checking
   * a lot!
   * if !strat->sbaOrder == 0
   * We are not subdividing lower and current index
   * due to the fact that we are using the induced
   * Schreyer order
   *
   * nevertheless, this different behaviour is
   * taken care of by is_sigsafe
   * => one reduction procedure can be used for
   * both, the incremental and the non-incremental
   * attempt!
   * --------------------------------------------
   *
   *********************************************/
  //printf("COMPARE IDX: %ld -- %ld\n",idx,strat->currIdx);
  if (!PW->is_sigsafe)
  {
    poly sigMult = pCopy(PW->sig);   // copy signature of reducer
//#if 1
#ifdef DEBUGF5
    printf("IN KSREDUCEPOLYSIG: \n");
    pWrite(pHead(f1));
    pWrite(pHead(f2));
    pWrite(sigMult);
    printf("--------------\n");
#endif
    p_ExpVectorAddSub(sigMult,PR->GetLmCurrRing(),PW->GetLmCurrRing(),currRing);
//#if 1
#ifdef DEBUGF5
    printf("------------------- IN KSREDUCEPOLYSIG: --------------------\n");
    pWrite(pHead(f1));
    pWrite(pHead(f2));
    pWrite(sigMult);
    pWrite(PR->sig);
    printf("--------------\n");
#endif
    int sigSafe = p_LmCmp(PR->sig,sigMult,currRing);
    // now we can delete the copied polynomial data used for checking for
    // sig-safeness of the reduction step
//#if 1
#ifdef DEBUGF5
    printf("%d -- %d sig\n",sigSafe,PW->is_sigsafe);

#endif
    //pDelete(&f1);
    pDelete(&sigMult);
    // go on with the computations only if the signature of p2 is greater than the
    // signature of fm*p1
    if(sigSafe != 1)
    {
      PR->is_redundant = TRUE;
      return 3;
    }
    //PW->is_sigsafe  = TRUE;
  }
  PR->is_redundant = FALSE;
  poly p1 = PR->GetLmTailRing();   // p2 | p1
  poly p2 = PW->GetLmTailRing();   // i.e. will reduce p1 with p2; lm = LT(p1) / LM(p2)
  poly t2 = pNext(p2), lm = p1;    // t2 = p2 - LT(p2); really compute P = LC(p2)*p1 - LT(p1)/LM(p2)*p2
  assume(p1 != NULL && p2 != NULL);// Attention, we have rings and there LC(p2) and LC(p1) are special
  p_CheckPolyRing(p1, tailRing);
  p_CheckPolyRing(p2, tailRing);

  pAssume1(p2 != NULL && p1 != NULL &&
      p_DivisibleBy(p2,  p1, tailRing));

  pAssume1(p_GetComp(p1, tailRing) == p_GetComp(p2, tailRing) ||
      (p_GetComp(p2, tailRing) == 0 &&
       p_MaxComp(pNext(p2),tailRing) == 0));

#ifdef HAVE_PLURAL
  if (rIsPluralRing(currRing))
  {
    // for the time being: we know currRing==strat->tailRing
    // no exp-bound checking needed
    // (only needed if exp-bound(tailring)<exp-b(currRing))
    if (PR->bucket!=NULL)  nc_kBucketPolyRed_Z(PR->bucket, p2,coef);
    else
    {
      poly _p = (PR->t_p != NULL ? PR->t_p : PR->p);
      assume(_p != NULL);
      nc_PolyPolyRed(_p, p2, coef, currRing);
      if (PR->t_p!=NULL) PR->t_p=_p; else PR->p=_p;
      PR->pLength=0; // usaully not used, GetpLength re-comoutes it if needed
    }
    return 0;
  }
#endif

  if (t2==NULL)           // Divisor is just one term, therefore it will
  {                       // just cancel the leading term
    PR->LmDeleteAndIter();
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
    return 0;
  }

  p_ExpVectorSub(lm, p2, tailRing); // Calculate the Monomial we must multiply to p2

  if (tailRing != currRing)
  {
    // check that reduction does not violate exp bound
    while (PW->max_exp != NULL && !p_LmExpVectorAddIsOk(lm, PW->max_exp, tailRing))
    {
      // undo changes of lm
      p_ExpVectorAdd(lm, p2, tailRing);
      if (strat == NULL) return 2;
      if (! kStratChangeTailRing(strat, PR, PW)) return -1;
      tailRing = strat->tailRing;
      p1 = PR->GetLmTailRing();
      p2 = PW->GetLmTailRing();
      t2 = pNext(p2);
      lm = p1;
      p_ExpVectorSub(lm, p2, tailRing);
      ret = 1;
    }
  }

  // take care of coef buisness
  if (! n_IsOne(pGetCoeff(p2), tailRing->cf))
  {
    number bn = pGetCoeff(lm);
    number an = pGetCoeff(p2);
    int ct = ksCheckCoeff(&an, &bn, tailRing->cf);    // Calculate special LC
    p_SetCoeff(lm, bn, tailRing);
    if ((ct == 0) || (ct == 2))
      PR->Tail_Mult_nn(an);
    if (coef != NULL) *coef = an;
    else n_Delete(&an, tailRing->cf);
  }
  else
  {
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
  }


  // and finally,
  PR->Tail_Minus_mm_Mult_qq(lm, t2, PW->GetpLength() - 1, spNoether);
  assume(PW->GetpLength() == pLength(PW->p != NULL ? PW->p : PW->t_p));
  PR->LmDeleteAndIter();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if ( (currRing->isLPring) && (!strat->homog) )
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif

#if defined(KDEBUG) && defined(TEST_OPT_DEBUG_RED)
  if (TEST_OPT_DEBUG)
  {
    Print(" to: "); PR->wrp(); Print("\n");
  }
#endif
  return ret;
}

int ksReducePolySigRing(LObject* PR,
                 TObject* PW,
                 long /*idx*/,
                 poly spNoether,
                 number *coef,
                 kStrategy strat)
{
#ifdef KDEBUG
  red_count++;
#ifdef TEST_OPT_DEBUG_RED
  if (TEST_OPT_DEBUG)
  {
    Print("Red %d:", red_count); PR->wrp(); Print(" with:");
    PW->wrp();
  }
#endif
#endif
  int ret = 0;
  ring tailRing = PR->tailRing;
  kTest_L(PR);
  kTest_T(PW);

  // signature-based stuff:
  // checking for sig-safeness first
  // NOTE: This has to be done in the current ring
  //
  /**********************************************
   *
   * TODO:
   * --------------------------------------------
   * if strat->sbaOrder == 1
   * Since we are subdividing lower index and
   * current index reductions it is enough to
   * look at the polynomial part of the signature
   * for a check. This should speed-up checking
   * a lot!
   * if !strat->sbaOrder == 0
   * We are not subdividing lower and current index
   * due to the fact that we are using the induced
   * Schreyer order
   *
   * nevertheless, this different behaviour is
   * taken care of by is_sigsafe
   * => one reduction procedure can be used for
   * both, the incremental and the non-incremental
   * attempt!
   * --------------------------------------------
   *
   *********************************************/
  //printf("COMPARE IDX: %ld -- %ld\n",idx,strat->currIdx);
  if (!PW->is_sigsafe)
  {
    poly sigMult = pCopy(PW->sig);   // copy signature of reducer
//#if 1
#ifdef DEBUGF5
    printf("IN KSREDUCEPOLYSIG: \n");
    pWrite(pHead(f1));
    pWrite(pHead(f2));
    pWrite(sigMult);
    printf("--------------\n");
#endif
    p_ExpVectorAddSub(sigMult,PR->GetLmCurrRing(),PW->GetLmCurrRing(),currRing);
    //I have also to set the leading coeficient for sigMult (in the case of rings)
    if(rField_is_Ring(currRing))
    {
      pSetCoeff(sigMult,nMult(nDiv(pGetCoeff(PR->p),pGetCoeff(PW->p)), pGetCoeff(sigMult)));
      if(nIsZero(pGetCoeff(sigMult)))
      {
        sigMult = NULL;
      }
    }
//#if 1
#ifdef DEBUGF5
    printf("------------------- IN KSREDUCEPOLYSIG: --------------------\n");
    pWrite(pHead(f1));
    pWrite(pHead(f2));
    pWrite(sigMult);
    pWrite(PR->sig);
    printf("--------------\n");
#endif
    int sigSafe;
    if(!rField_is_Ring(currRing))
      sigSafe = p_LmCmp(PR->sig,sigMult,currRing);
    // now we can delete the copied polynomial data used for checking for
    // sig-safeness of the reduction step
//#if 1
#ifdef DEBUGF5
    printf("%d -- %d sig\n",sigSafe,PW->is_sigsafe);

#endif
    if(rField_is_Ring(currRing))
    {
      // Set the sig
      poly origsig = pCopy(PR->sig);
      if(sigMult != NULL)
        PR->sig = pHead(pSub(PR->sig, sigMult));
      //The sigs have the same lm, have to substract
      //It may happen that now the signature is 0 (drop)
      if(PR->sig == NULL)
      {
        strat->sigdrop=TRUE;
      }
      else
      {
        if(pLtCmp(PR->sig,origsig) == 1)
        {
          // do not allow this reduction - it will increase it's signature
          // and the partially standard basis is just till the old sig, not the new one
          PR->is_redundant = TRUE;
          pDelete(&PR->sig);
          PR->sig = origsig;
          strat->blockred++;
          return 3;
        }
        if(pLtCmp(PR->sig,origsig) == -1)
        {
          strat->sigdrop=TRUE;
        }
      }
      pDelete(&origsig);
    }
    //pDelete(&f1);
    // go on with the computations only if the signature of p2 is greater than the
    // signature of fm*p1
    if(sigSafe != 1 && !rField_is_Ring(currRing))
    {
      PR->is_redundant = TRUE;
      return 3;
    }
    //PW->is_sigsafe  = TRUE;
  }
  PR->is_redundant = FALSE;
  poly p1 = PR->GetLmTailRing();   // p2 | p1
  poly p2 = PW->GetLmTailRing();   // i.e. will reduce p1 with p2; lm = LT(p1) / LM(p2)
  poly t2 = pNext(p2), lm = p1;    // t2 = p2 - LT(p2); really compute P = LC(p2)*p1 - LT(p1)/LM(p2)*p2
  assume(p1 != NULL && p2 != NULL);// Attention, we have rings and there LC(p2) and LC(p1) are special
  p_CheckPolyRing(p1, tailRing);
  p_CheckPolyRing(p2, tailRing);

  pAssume1(p2 != NULL && p1 != NULL &&
      p_DivisibleBy(p2,  p1, tailRing));

  pAssume1(p_GetComp(p1, tailRing) == p_GetComp(p2, tailRing) ||
      (p_GetComp(p2, tailRing) == 0 &&
       p_MaxComp(pNext(p2),tailRing) == 0));

#ifdef HAVE_PLURAL
  if (rIsPluralRing(currRing))
  {
    // for the time being: we know currRing==strat->tailRing
    // no exp-bound checking needed
    // (only needed if exp-bound(tailring)<exp-b(currRing))
    if (PR->bucket!=NULL)  nc_kBucketPolyRed_Z(PR->bucket, p2,coef);
    else
    {
      poly _p = (PR->t_p != NULL ? PR->t_p : PR->p);
      assume(_p != NULL);
      nc_PolyPolyRed(_p, p2, coef, currRing);
      if (PR->t_p!=NULL) PR->t_p=_p; else PR->p=_p;
      PR->pLength=0; // usaully not used, GetpLength re-comoutes it if needed
    }
    return 0;
  }
#endif

  if (t2==NULL)           // Divisor is just one term, therefore it will
  {                       // just cancel the leading term
    PR->LmDeleteAndIter();
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
    return 0;
  }

  p_ExpVectorSub(lm, p2, tailRing); // Calculate the Monomial we must multiply to p2

  if (tailRing != currRing)
  {
    // check that reduction does not violate exp bound
    while (PW->max_exp != NULL && !p_LmExpVectorAddIsOk(lm, PW->max_exp, tailRing))
    {
      // undo changes of lm
      p_ExpVectorAdd(lm, p2, tailRing);
      if (strat == NULL) return 2;
      if (! kStratChangeTailRing(strat, PR, PW)) return -1;
      tailRing = strat->tailRing;
      p1 = PR->GetLmTailRing();
      p2 = PW->GetLmTailRing();
      t2 = pNext(p2);
      lm = p1;
      p_ExpVectorSub(lm, p2, tailRing);
      ret = 1;
    }
  }
  // take care of coef buisness
  if(rField_is_Ring(currRing))
  {
    p_SetCoeff(lm, nDiv(pGetCoeff(lm),pGetCoeff(p2)), tailRing);
    if (coef != NULL) *coef = n_Init(1, tailRing->cf);
  }
  else
  {
    if (! n_IsOne(pGetCoeff(p2), tailRing->cf))
    {
      number bn = pGetCoeff(lm);
      number an = pGetCoeff(p2);
      int ct = ksCheckCoeff(&an, &bn, tailRing->cf);    // Calculate special LC
      p_SetCoeff(lm, bn, tailRing);
      if (((ct == 0) || (ct == 2)))
        PR->Tail_Mult_nn(an);
      if (coef != NULL) *coef = an;
      else n_Delete(&an, tailRing->cf);
    }
    else
    {
      if (coef != NULL) *coef = n_Init(1, tailRing->cf);
    }
  }

  // and finally,
  PR->Tail_Minus_mm_Mult_qq(lm, t2, PW->GetpLength() - 1, spNoether);
  assume(PW->GetpLength() == pLength(PW->p != NULL ? PW->p : PW->t_p));
  PR->LmDeleteAndIter();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if ( (currRing->isLPring) && (!strat->homog) )
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif
#if defined(KDEBUG) && defined(TEST_OPT_DEBUG_RED)
  if (TEST_OPT_DEBUG)
  {
    Print(" to: "); PR->wrp(); Print("\n");
  }
#endif
  return ret;
}

/***************************************************************
 *
 * Creates S-Poly of p1 and p2
 *
 *
 ***************************************************************/
void ksCreateSpoly(LObject* Pair,   poly spNoether,
                   int use_buckets, ring tailRing,
                   poly m1, poly m2, TObject** R)
{
#ifdef KDEBUG
  create_count++;
#endif
  kTest_L(Pair);
  poly p1 = Pair->p1;
  poly p2 = Pair->p2;
  Pair->tailRing = tailRing;

  assume(p1 != NULL);
  assume(p2 != NULL);
  assume(tailRing != NULL);

  poly a1 = pNext(p1), a2 = pNext(p2);
  number lc1 = pGetCoeff(p1), lc2 = pGetCoeff(p2);
  int co=0/*, ct = ksCheckCoeff(&lc1, &lc2, currRing->cf)*/; // gcd and zero divisors
  (void) ksCheckCoeff(&lc1, &lc2, currRing->cf);

  int l1=0, l2=0;

  if (currRing->pCompIndex >= 0)
  {
    if (__p_GetComp(p1, currRing)!=__p_GetComp(p2, currRing))
    {
      if (__p_GetComp(p1, currRing)==0)
      {
        co=1;
        p_SetCompP(p1,__p_GetComp(p2, currRing), currRing, tailRing);
      }
      else
      {
        co=2;
        p_SetCompP(p2, __p_GetComp(p1, currRing), currRing, tailRing);
      }
    }
  }

  // get m1 = LCM(LM(p1), LM(p2))/LM(p1)
  //     m2 = LCM(LM(p1), LM(p2))/LM(p2)
  if (m1 == NULL)
    k_GetLeadTerms(p1, p2, currRing, m1, m2, tailRing);

  pSetCoeff0(m1, lc2);
  pSetCoeff0(m2, lc1);  // and now, m1 * LT(p1) == m2 * LT(p2)

  if (R != NULL)
  {
    if (Pair->i_r1 == -1)
    {
      l1 = pLength(p1) - 1;
    }
    else
    {
      l1 = (R[Pair->i_r1])->GetpLength() - 1;
    }
    if ((Pair->i_r2 == -1)||(R[Pair->i_r2]==NULL))
    {
      l2 = pLength(p2) - 1;
    }
    else
    {
      l2 = (R[Pair->i_r2])->GetpLength() - 1;
    }
  }

  // get m2 * a2
  if (spNoether != NULL)
  {
    l2 = -1;
    a2 = tailRing->p_Procs->pp_Mult_mm_Noether(a2, m2, spNoether, l2, tailRing);
    assume(l2 == pLength(a2));
  }
  else
    a2 = tailRing->p_Procs->pp_Mult_mm(a2, m2, tailRing);
#ifdef HAVE_RINGS
  if (!(rField_is_Domain(currRing))) l2 = pLength(a2);
#endif

  Pair->SetLmTail(m2, a2, l2, use_buckets, tailRing);

  // get m2*a2 - m1*a1
  Pair->Tail_Minus_mm_Mult_qq(m1, a1, l1, spNoether);

  // Clean-up time
  Pair->LmDeleteAndIter();
  p_LmDelete(m1, tailRing);

  if (co != 0)
  {
    if (co==1)
    {
      p_SetCompP(p1,0, currRing, tailRing);
    }
    else
    {
      p_SetCompP(p2,0, currRing, tailRing);
    }
  }

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if (currRing->isLPring)
  {
    // assume? h->p in currRing
    Pair->GetP();
    poly qq = p_Shrink(Pair->p, currRing->isLPring, currRing);
    Pair->Clear(); // does the right things
    Pair->p = qq;
    Pair->t_p = NULL;
    Pair->SetShortExpVector();
  }
#endif

}

int ksReducePolyTail(LObject* PR, TObject* PW, poly Current, poly spNoether)
{
  BOOLEAN ret;
  number coef;
  poly Lp =     PR->GetLmCurrRing();
  poly Save =   PW->GetLmCurrRing();

  kTest_L(PR);
  kTest_T(PW);
  pAssume(pIsMonomOf(Lp, Current));

  assume(Lp != NULL && Current != NULL && pNext(Current) != NULL);
  assume(PR->bucket == NULL);

  LObject Red(pNext(Current), PR->tailRing);
  TObject With(PW, Lp == Save);

  pAssume(!pHaveCommonMonoms(Red.p, With.p));
  ret = ksReducePoly(&Red, &With, spNoether, &coef);

  if (!ret)
  {
    if (! n_IsOne(coef, currRing->cf))
    {
      pNext(Current) = NULL;
      if (Current == PR->p && PR->t_p != NULL)
        pNext(PR->t_p) = NULL;
      PR->Mult_nn(coef);
    }

    n_Delete(&coef, currRing->cf);
    pNext(Current) = Red.GetLmTailRing();
    if (Current == PR->p && PR->t_p != NULL)
      pNext(PR->t_p) = pNext(Current);
  }

  if (Lp == Save)
    With.Delete();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if (currRing->isLPring)
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif

  return ret;
}

int ksReducePolyTailBound(LObject* PR, TObject* PW, int bound, poly Current, poly spNoether)
{
  BOOLEAN ret;
  number coef;
  poly Lp =     PR->GetLmCurrRing();
  poly Save =   PW->GetLmCurrRing();

  kTest_L(PR);
  kTest_T(PW);
  pAssume(pIsMonomOf(Lp, Current));

  assume(Lp != NULL && Current != NULL && pNext(Current) != NULL);
  assume(PR->bucket == NULL);

  LObject Red(pNext(Current), PR->tailRing);
  TObject With(PW, Lp == Save);

  pAssume(!pHaveCommonMonoms(Red.p, With.p));
  ret = ksReducePolyBound(&Red, &With,bound, spNoether, &coef);

  if (!ret)
  {
    if (! n_IsOne(coef, currRing))
    {
      pNext(Current) = NULL;
      if (Current == PR->p && PR->t_p != NULL)
        pNext(PR->t_p) = NULL;
      PR->Mult_nn(coef);
    }

    n_Delete(&coef, currRing);
    pNext(Current) = Red.GetLmTailRing();
    if (Current == PR->p && PR->t_p != NULL)
      pNext(PR->t_p) = pNext(Current);
  }

  if (Lp == Save)
    With.Delete();

  // the following is commented out: shrinking
#ifdef HAVE_SHIFTBBA_NONEXISTENT
  if (currRing->isLPring)
  {
    // assume? h->p in currRing
    PR->GetP();
    poly qq = p_Shrink(PR->p, currRing->isLPring, currRing);
    PR->Clear(); // does the right things
    PR->p = qq;
    PR->t_p = NULL;
    PR->SetShortExpVector();
  }
#endif

  return ret;
}

/***************************************************************
 *
 * Auxillary Routines
 *
 *
 ***************************************************************/

/*2
* creates the leading term of the S-polynomial of p1 and p2
* do not destroy p1 and p2
* remarks:
*   1. the coefficient is 0 (p_Init)
*   1. a) in the case of coefficient ring, the coefficient is calculated
*   2. pNext is undefined
*/
//static void bbb() { int i=0; }
poly ksCreateShortSpoly(poly p1, poly p2, ring tailRing)
{
  poly a1 = pNext(p1), a2 = pNext(p2);
  long c1=p_GetComp(p1, currRing),c2=p_GetComp(p2, currRing);
  long c;
  poly m1,m2;
  number t1 = NULL,t2 = NULL;
  int cm,i;
  BOOLEAN equal;

#ifdef HAVE_RINGS
  BOOLEAN is_Ring=rField_is_Ring(currRing);
  number lc1 = pGetCoeff(p1), lc2 = pGetCoeff(p2);
  if (is_Ring)
  {
    ksCheckCoeff(&lc1, &lc2, currRing->cf); // gcd and zero divisors
    if (a1 != NULL) t2 = nMult(pGetCoeff(a1),lc2);
    if (a2 != NULL) t1 = nMult(pGetCoeff(a2),lc1);
    while (a1 != NULL && nIsZero(t2))
    {
      pIter(a1);
      nDelete(&t2);
      if (a1 != NULL) t2 = nMult(pGetCoeff(a1),lc2);
    }
    while (a2 != NULL && nIsZero(t1))
    {
      pIter(a2);
      nDelete(&t1);
      if (a2 != NULL) t1 = nMult(pGetCoeff(a2),lc1);
    }
  }
#endif

  if (a1==NULL)
  {
    if(a2!=NULL)
    {
      m2=p_Init(currRing);
x2:
      for (i = (currRing->N); i; i--)
      {
        c = p_GetExpDiff(p1, p2,i, currRing);
        if (c>0)
        {
          p_SetExp(m2,i,(c+p_GetExp(a2,i,tailRing)),currRing);
        }
        else
        {
          p_SetExp(m2,i,p_GetExp(a2,i,tailRing),currRing);
        }
      }
      if ((c1==c2)||(c2!=0))
      {
        p_SetComp(m2,p_GetComp(a2,tailRing), currRing);
      }
      else
      {
        p_SetComp(m2,c1,currRing);
      }
      p_Setm(m2, currRing);
#ifdef HAVE_RINGS
      if (is_Ring)
      {
          nDelete(&lc1);
          nDelete(&lc2);
          nDelete(&t2);
          pSetCoeff0(m2, t1);
      }
#endif
      return m2;
    }
    else
    {
#ifdef HAVE_RINGS
      if (is_Ring)
      {
        nDelete(&lc1);
        nDelete(&lc2);
        nDelete(&t1);
        nDelete(&t2);
      }
#endif
      return NULL;
    }
  }
  if (a2==NULL)
  {
    m1=p_Init(currRing);
x1:
    for (i = (currRing->N); i; i--)
    {
      c = p_GetExpDiff(p2, p1,i,currRing);
      if (c>0)
      {
        p_SetExp(m1,i,(c+p_GetExp(a1,i, tailRing)),currRing);
      }
      else
      {
        p_SetExp(m1,i,p_GetExp(a1,i, tailRing), currRing);
      }
    }
    if ((c1==c2)||(c1!=0))
    {
      p_SetComp(m1,p_GetComp(a1,tailRing),currRing);
    }
    else
    {
      p_SetComp(m1,c2,currRing);
    }
    p_Setm(m1, currRing);
#ifdef HAVE_RINGS
    if (is_Ring)
    {
      pSetCoeff0(m1, t2);
      nDelete(&lc1);
      nDelete(&lc2);
      nDelete(&t1);
    }
#endif
    return m1;
  }
  m1 = p_Init(currRing);
  m2 = p_Init(currRing);
  loop
  {
    for (i = (currRing->N); i; i--)
    {
      c = p_GetExpDiff(p1, p2,i,currRing);
      if (c > 0)
      {
        p_SetExp(m2,i,(c+p_GetExp(a2,i,tailRing)), currRing);
        p_SetExp(m1,i,p_GetExp(a1,i, tailRing), currRing);
      }
      else
      {
        p_SetExp(m1,i,(p_GetExp(a1,i,tailRing)-c), currRing);
        p_SetExp(m2,i,p_GetExp(a2,i, tailRing), currRing);
      }
    }
    if(c1==c2)
    {
      p_SetComp(m1,p_GetComp(a1, tailRing), currRing);
      p_SetComp(m2,p_GetComp(a2, tailRing), currRing);
    }
    else
    {
      if(c1!=0)
      {
        p_SetComp(m1,p_GetComp(a1, tailRing), currRing);
        p_SetComp(m2,c1, currRing);
      }
      else
      {
        p_SetComp(m2,p_GetComp(a2, tailRing), currRing);
        p_SetComp(m1,c2, currRing);
      }
    }
    p_Setm(m1,currRing);
    p_Setm(m2,currRing);
    cm = p_LmCmp(m1, m2,currRing);
    if (cm!=0)
    {
      if(cm==1)
      {
        p_LmFree(m2,currRing);
#ifdef HAVE_RINGS
        if (is_Ring)
        {
          pSetCoeff0(m1, t2);
          nDelete(&lc1);
          nDelete(&lc2);
          nDelete(&t1);
        }
#endif
        return m1;
      }
      else
      {
        p_LmFree(m1,currRing);
#ifdef HAVE_RINGS
        if (is_Ring)
        {
          pSetCoeff0(m2, t1);
          nDelete(&lc1);
          nDelete(&lc2);
          nDelete(&t2);
        }
#endif
        return m2;
      }
    }
#ifdef HAVE_RINGS
    if (is_Ring)
    {
      equal = nEqual(t1,t2);
    }
    else
#endif
    {
      t1 = nMult(pGetCoeff(a2),pGetCoeff(p1));
      t2 = nMult(pGetCoeff(a1),pGetCoeff(p2));
      equal = nEqual(t1,t2);
      nDelete(&t2);
      nDelete(&t1);
    }
    if (!equal)
    {
      p_LmFree(m2,currRing);
#ifdef HAVE_RINGS
      if (is_Ring)
      {
          pSetCoeff0(m1, nSub(t1, t2));
          nDelete(&lc1);
          nDelete(&lc2);
          nDelete(&t1);
          nDelete(&t2);
      }
#endif
      return m1;
    }
    pIter(a1);
    pIter(a2);
#ifdef HAVE_RINGS
    if (is_Ring)
    {
      if (a2 != NULL)
      {
        nDelete(&t1);
        t1 = nMult(pGetCoeff(a2),lc1);
      }
      if (a1 != NULL)
      {
        nDelete(&t2);
        t2 = nMult(pGetCoeff(a1),lc2);
      }
      while ((a1 != NULL) && nIsZero(t2))
      {
        pIter(a1);
        if (a1 != NULL)
        {
          nDelete(&t2);
          t2 = nMult(pGetCoeff(a1),lc2);
        }
      }
      while ((a2 != NULL) && nIsZero(t1))
      {
        pIter(a2);
        if (a2 != NULL)
        {
          nDelete(&t1);
          t1 = nMult(pGetCoeff(a2),lc1);
        }
      }
    }
#endif
    if (a2==NULL)
    {
      p_LmFree(m2,currRing);
      if (a1==NULL)
      {
#ifdef HAVE_RINGS
        if (is_Ring)
        {
          nDelete(&lc1);
          nDelete(&lc2);
          nDelete(&t1);
          nDelete(&t2);
        }
#endif
        p_LmFree(m1,currRing);
        return NULL;
      }
      goto x1;
    }
    if (a1==NULL)
    {
      p_LmFree(m1,currRing);
      goto x2;
    }
  }
}