hdegree.cc

Go to the documentation of this file.

/****************************************
*  Computer Algebra System SINGULAR     *
****************************************/
/*
*  ABSTRACT -  dimension, multiplicity, HC, kbase
*/
 
#include "kernel/mod2.h"
 
#include "misc/intvec.h"
#include "coeffs/numbers.h"
 
#include "kernel/structs.h"
#include "kernel/ideals.h"
#include "kernel/polys.h"
 
#include "kernel/combinatorics/hutil.h"
#include "kernel/combinatorics/hilb.h"
#include "kernel/combinatorics/stairc.h"
#include "reporter/reporter.h"
 
#ifdef HAVE_SHIFTBBA
#include <vector>
#include "misc/options.h"
#endif
 
VAR int  hCo, hMu2;
VAR long hMu;
VAR omBin indlist_bin = omGetSpecBin(sizeof(indlist));
 
/*0 implementation*/
 
// dimension
 
void hDimSolve(scmon pure, int Npure, scfmon rad, int Nrad,
 varset var, int Nvar)
{
  int  dn, iv, rad0, b, c, x;
  scmon pn;
  scfmon rn;
  if (Nrad < 2)
  {
    dn = Npure + Nrad;
    if (dn < hCo)
      hCo = dn;
    return;
  }
  if (Npure+1 >= hCo)
    return;
  iv = Nvar;
  while(pure[var[iv]]) iv--;
  hStepR(rad, Nrad, var, iv, &rad0);
  if (rad0!=0)
  {
    iv--;
    if (rad0 < Nrad)
    {
      pn = hGetpure(pure);
      rn = hGetmem(Nrad, rad, radmem[iv]);
      hDimSolve(pn, Npure + 1, rn, rad0, var, iv);
      b = rad0;
      c = Nrad;
      hElimR(rn, &rad0, b, c, var, iv);
      hPure(rn, b, &c, var, iv, pn, &x);
      hLex2R(rn, rad0, b, c, var, iv, hwork);
      rad0 += (c - b);
      hDimSolve(pn, Npure + x, rn, rad0, var, iv);
    }
    else
    {
      hDimSolve(pure, Npure, rad, Nrad, var, iv);
    }
  }
  else
    hCo = Npure + 1;
}
 
int  scDimInt(ideal S, ideal Q)
{
  id_Test(S, currRing);
  if( Q!=NULL ) id_Test(Q, currRing);
 
  int  mc;
  hexist = hInit(S, Q, &hNexist);
  if (!hNexist)
    return (currRing->N);
  hwork = (scfmon)omAlloc(hNexist * sizeof(scmon));
  hvar = (varset)omAlloc(((currRing->N) + 1) * sizeof(int));
  hpure = (scmon)omAlloc((1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  mc = hisModule;
  if (!mc)
  {
    hrad = hexist;
    hNrad = hNexist;
  }
  else
    hrad = (scfmon)omAlloc(hNexist * sizeof(scmon));
  radmem = hCreate((currRing->N) - 1);
  hCo = (currRing->N) + 1;
  loop
  {
    if (mc)
      hComp(hexist, hNexist, mc, hrad, &hNrad);
    if (hNrad)
    {
      hNvar = (currRing->N);
      hRadical(hrad, &hNrad, hNvar);
      hSupp(hrad, hNrad, hvar, &hNvar);
      if (hNvar)
      {
        memset(hpure, 0, ((currRing->N) + 1) * sizeof(int));
        hPure(hrad, 0, &hNrad, hvar, hNvar, hpure, &hNpure);
        hLexR(hrad, hNrad, hvar, hNvar);
        hDimSolve(hpure, hNpure, hrad, hNrad, hvar, hNvar);
      }
    }
    else
    {
      hCo = 0;
      break;
    }
    mc--;
    if (mc <= 0)
      break;
  }
  hKill(radmem, (currRing->N) - 1);
  omFreeSize((ADDRESS)hpure, (1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  omFreeSize((ADDRESS)hvar, ((currRing->N) + 1) * sizeof(int));
  omFreeSize((ADDRESS)hwork, hNexist * sizeof(scmon));
  hDelete(hexist, hNexist);
  if (hisModule)
    omFreeSize((ADDRESS)hrad, hNexist * sizeof(scmon));
  return (currRing->N) - hCo;
}
 
int  scDimIntRing(ideal vid, ideal Q)
{
#ifdef HAVE_RINGS
  if (rField_is_Ring(currRing))
  {
    int i = idPosConstant(vid);
    if ((i != -1) && (n_IsUnit(pGetCoeff(vid->m[i]),currRing->cf)))
    { /* ideal v contains unit; dim = -1 */
      return(-1);
    }
    ideal vv = id_Head(vid,currRing);
    idSkipZeroes(vv);
    i = idPosConstant(vid);
    int d;
    if(i == -1)
    {
      d = scDimInt(vv, Q);
      if(rField_is_Z(currRing))
        d++;
    }
    else
    {
      if(n_IsUnit(pGetCoeff(vv->m[i]),currRing->cf))
        d = -1;
      else
        d = scDimInt(vv, Q);
    }
    //Anne's Idea for std(4,2x) = 0 bug
    int dcurr = d;
    for(unsigned ii=0;ii<(unsigned)IDELEMS(vv);ii++)
    {
      if(vv->m[ii] != NULL && !n_IsUnit(pGetCoeff(vv->m[ii]),currRing->cf))
      {
        ideal vc = idCopy(vv);
        poly c = pInit();
        pSetCoeff0(c,nCopy(pGetCoeff(vv->m[ii])));
        idInsertPoly(vc,c);
        idSkipZeroes(vc);
        for(unsigned jj = 0;jj<(unsigned)IDELEMS(vc)-1;jj++)
        {
          if((vc->m[jj]!=NULL)
          && (n_DivBy(pGetCoeff(vc->m[jj]),pGetCoeff(c),currRing->cf)))
          {
            pDelete(&vc->m[jj]);
          }
        }
        idSkipZeroes(vc);
        i = idPosConstant(vc);
        if (i != -1) pDelete(&vc->m[i]);
        dcurr = scDimInt(vc, Q);
        // the following assumes the ground rings to be either zero- or one-dimensional
        if((i==-1) && rField_is_Z(currRing))
        {
          // should also be activated for other euclidean domains as groundfield
          dcurr++;
        }
        idDelete(&vc);
      }
      if(dcurr > d)
          d = dcurr;
    }
    idDelete(&vv);
    return d;
  }
#endif
  return scDimInt(vid,Q);
}
 
// independent set
STATIC_VAR scmon hInd;
 
static void hIndSolve(scmon pure, int Npure, scfmon rad, int Nrad,
 varset var, int Nvar)
{
  int  dn, iv, rad0, b, c, x;
  scmon pn;
  scfmon rn;
  if (Nrad < 2)
  {
    dn = Npure + Nrad;
    if (dn < hCo)
    {
      hCo = dn;
      for (iv=(currRing->N); iv; iv--)
      {
        if (pure[iv])
          hInd[iv] = 0;
        else
          hInd[iv] = 1;
      }
      if (Nrad)
      {
        pn = *rad;
        iv = Nvar;
        loop
        {
          x = var[iv];
          if (pn[x])
          {
            hInd[x] = 0;
            break;
          }
          iv--;
        }
      }
    }
    return;
  }
  if (Npure+1 >= hCo)
    return;
  iv = Nvar;
  while(pure[var[iv]]) iv--;
  hStepR(rad, Nrad, var, iv, &rad0);
  if (rad0)
  {
    iv--;
    if (rad0 < Nrad)
    {
      pn = hGetpure(pure);
      rn = hGetmem(Nrad, rad, radmem[iv]);
      pn[var[iv + 1]] = 1;
      hIndSolve(pn, Npure + 1, rn, rad0, var, iv);
      pn[var[iv + 1]] = 0;
      b = rad0;
      c = Nrad;
      hElimR(rn, &rad0, b, c, var, iv);
      hPure(rn, b, &c, var, iv, pn, &x);
      hLex2R(rn, rad0, b, c, var, iv, hwork);
      rad0 += (c - b);
      hIndSolve(pn, Npure + x, rn, rad0, var, iv);
    }
    else
    {
      hIndSolve(pure, Npure, rad, Nrad, var, iv);
    }
  }
  else
  {
    hCo = Npure + 1;
    for (x=(currRing->N); x; x--)
    {
      if (pure[x])
        hInd[x] = 0;
      else
        hInd[x] = 1;
    }
    hInd[var[iv]] = 0;
  }
}
 
intvec * scIndIntvec(ideal S, ideal Q)
{
  id_Test(S, currRing);
  if( Q!=NULL ) id_Test(Q, currRing);
 
  intvec *Set=new intvec((currRing->N));
  int  mc,i;
  hexist = hInit(S, Q, &hNexist);
  if (hNexist==0)
  {
    for(i=0; i<(currRing->N); i++)
      (*Set)[i]=1;
    return Set;
  }
  hwork = (scfmon)omAlloc(hNexist * sizeof(scmon));
  hvar = (varset)omAlloc(((currRing->N) + 1) * sizeof(int));
  hpure = (scmon)omAlloc((1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  hInd = (scmon)omAlloc0((1 + (currRing->N)) * sizeof(int));
  mc = hisModule;
  if (mc==0)
  {
    hrad = hexist;
    hNrad = hNexist;
  }
  else
    hrad = (scfmon)omAlloc(hNexist * sizeof(scmon));
  radmem = hCreate((currRing->N) - 1);
  hCo = (currRing->N) + 1;
  loop
  {
    if (mc!=0)
      hComp(hexist, hNexist, mc, hrad, &hNrad);
    if (hNrad!=0)
    {
      hNvar = (currRing->N);
      hRadical(hrad, &hNrad, hNvar);
      hSupp(hrad, hNrad, hvar, &hNvar);
      if (hNvar!=0)
      {
        memset(hpure, 0, ((currRing->N) + 1) * sizeof(int));
        hPure(hrad, 0, &hNrad, hvar, hNvar, hpure, &hNpure);
        hLexR(hrad, hNrad, hvar, hNvar);
        hIndSolve(hpure, hNpure, hrad, hNrad, hvar, hNvar);
      }
    }
    else
    {
      hCo = 0;
      break;
    }
    mc--;
    if (mc <= 0)
      break;
  }
  for(i=0; i<(currRing->N); i++)
    (*Set)[i] = hInd[i+1];
  hKill(radmem, (currRing->N) - 1);
  omFreeSize((ADDRESS)hpure, (1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  omFreeSize((ADDRESS)hInd, (1 + (currRing->N)) * sizeof(int));
  omFreeSize((ADDRESS)hvar, ((currRing->N) + 1) * sizeof(int));
  omFreeSize((ADDRESS)hwork, hNexist * sizeof(scmon));
  hDelete(hexist, hNexist);
  if (hisModule)
    omFreeSize((ADDRESS)hrad, hNexist * sizeof(scmon));
  return Set;
}
 
VAR indset ISet, JSet;
 
static BOOLEAN hNotZero(scfmon rad, int Nrad, varset var, int Nvar)
{
  int  k1, i;
  k1 = var[Nvar];
  i = 0;
  loop
  {
    if (rad[i][k1]==0)
      return FALSE;
    i++;
    if (i == Nrad)
      return TRUE;
  }
}
 
static void hIndep(scmon pure)
{
  int iv;
  intvec *Set;
 
  Set = ISet->set = new intvec((currRing->N));
  for (iv=(currRing->N); iv!=0 ; iv--)
  {
    (*Set)[iv-1] = (pure[iv]==0);
  }
  ISet = ISet->nx = (indset)omAlloc0Bin(indlist_bin);
  hMu++;
}
 
void hIndMult(scmon pure, int Npure, scfmon rad, int Nrad,
 varset var, int Nvar)
{
  int  dn, iv, rad0, b, c, x;
  scmon pn;
  scfmon rn;
  if (Nrad < 2)
  {
    dn = Npure + Nrad;
    if (dn == hCo)
    {
      if (Nrad==0)
        hIndep(pure);
      else
      {
        pn = *rad;
        for (iv = Nvar; iv!=0; iv--)
        {
          x = var[iv];
          if (pn[x])
          {
            pure[x] = 1;
            hIndep(pure);
            pure[x] = 0;
          }
        }
      }
    }
    return;
  }
  iv = Nvar;
  dn = Npure+1;
  if (dn >= hCo)
  {
    if (dn > hCo)
      return;
    loop
    {
      if(!pure[var[iv]])
      {
        if(hNotZero(rad, Nrad, var, iv))
        {
          pure[var[iv]] = 1;
          hIndep(pure);
          pure[var[iv]] = 0;
        }
      }
      iv--;
      if (!iv)
        return;
    }
  }
  while(pure[var[iv]]) iv--;
  hStepR(rad, Nrad, var, iv, &rad0);
  iv--;
  if (rad0 < Nrad)
  {
    pn = hGetpure(pure);
    rn = hGetmem(Nrad, rad, radmem[iv]);
    pn[var[iv + 1]] = 1;
    hIndMult(pn, Npure + 1, rn, rad0, var, iv);
    pn[var[iv + 1]] = 0;
    b = rad0;
    c = Nrad;
    hElimR(rn, &rad0, b, c, var, iv);
    hPure(rn, b, &c, var, iv, pn, &x);
    hLex2R(rn, rad0, b, c, var, iv, hwork);
    rad0 += (c - b);
    hIndMult(pn, Npure + x, rn, rad0, var, iv);
  }
  else
  {
    hIndMult(pure, Npure, rad, Nrad, var, iv);
  }
}
 
/*3
* consider indset x := !pure
* (for all i) (if(sm(i) > x) return FALSE)
* else return TRUE
*/
static BOOLEAN hCheck1(indset sm, scmon pure)
{
  int iv;
  intvec *Set;
  while (sm->nx != NULL)
  {
    Set = sm->set;
    iv=(currRing->N);
    loop
    {
      if (((*Set)[iv-1] == 0) && (pure[iv] == 0))
        break;
      iv--;
      if (iv == 0)
        return FALSE;
    }
    sm = sm->nx;
  }
  return TRUE;
}
 
/*3
* consider indset x := !pure
* (for all i) if(x > sm(i)) delete sm(i))
* return (place for x)
*/
static indset hCheck2(indset sm, scmon pure)
{
  int iv;
  intvec *Set;
  indset be, a1 = NULL;
  while (sm->nx != NULL)
  {
    Set = sm->set;
    iv=(currRing->N);
    loop
    {
      if ((pure[iv] == 1) && ((*Set)[iv-1] == 1))
        break;
      iv--;
      if (iv == 0)
      {
        if (a1 == NULL)
        {
          a1 = sm;
        }
        else
        {
          hMu2--;
          be->nx = sm->nx;
          delete Set;
          omFreeBin((ADDRESS)sm, indlist_bin);
          sm = be;
        }
        break;
      }
    }
    be = sm;
    sm = sm->nx;
  }
  if (a1 != NULL)
  {
    return a1;
  }
  else
  {
    hMu2++;
    sm->set = new intvec((currRing->N));
    sm->nx = (indset)omAlloc0Bin(indlist_bin);
    return sm;
  }
}
 
/*2
*  definition x >= y
*      x(i) == 0 => y(i) == 0
*      > ex. j with x(j) == 1 and y(j) == 0
*/
static void hCheckIndep(scmon pure)
{
  intvec *Set;
  indset res;
  int iv;
  if (hCheck1(ISet, pure))
  {
    if (hCheck1(JSet, pure))
    {
      res = hCheck2(JSet,pure);
      if (res == NULL)
        return;
      Set = res->set;
      for (iv=(currRing->N); iv; iv--)
      {
        (*Set)[iv-1] = (pure[iv]==0);
      }
    }
  }
}
 
void hIndAllMult(scmon pure, int Npure, scfmon rad, int Nrad,
 varset var, int Nvar)
{
  int  dn, iv, rad0, b, c, x;
  scmon pn;
  scfmon rn;
  if (Nrad < 2)
  {
    dn = Npure + Nrad;
    if (dn > hCo)
    {
      if (!Nrad)
        hCheckIndep(pure);
      else
      {
        pn = *rad;
        for (iv = Nvar; iv; iv--)
        {
          x = var[iv];
          if (pn[x])
          {
            pure[x] = 1;
            hCheckIndep(pure);
            pure[x] = 0;
          }
        }
      }
    }
    return;
  }
  iv = Nvar;
  while(pure[var[iv]]) iv--;
  hStepR(rad, Nrad, var, iv, &rad0);
  iv--;
  if (rad0 < Nrad)
  {
    pn = hGetpure(pure);
    rn = hGetmem(Nrad, rad, radmem[iv]);
    pn[var[iv + 1]] = 1;
    hIndAllMult(pn, Npure + 1, rn, rad0, var, iv);
    pn[var[iv + 1]] = 0;
    b = rad0;
    c = Nrad;
    hElimR(rn, &rad0, b, c, var, iv);
    hPure(rn, b, &c, var, iv, pn, &x);
    hLex2R(rn, rad0, b, c, var, iv, hwork);
    rad0 += (c - b);
    hIndAllMult(pn, Npure + x, rn, rad0, var, iv);
  }
  else
  {
    hIndAllMult(pure, Npure, rad, Nrad, var, iv);
  }
}
 
// multiplicity
 
static long hZeroMult(scmon pure, scfmon stc, int Nstc, varset var, int Nvar)
{
  int  iv = Nvar -1, a, a0, a1, b, i;
  long sum;
  int  x, x0;
  scmon pn;
  scfmon sn;
  if (!iv)
    return pure[var[1]];
  else if (!Nstc)
  {
    sum = 1;
    for (i = Nvar; i; i--)
      sum *= pure[var[i]];
    return sum;
  }
  x = a = 0;
  pn = hGetpure(pure);
  sn = hGetmem(Nstc, stc, stcmem[iv]);
  hStepS(sn, Nstc, var, Nvar, &a, &x);
  if (a == Nstc)
  {
    #if  SIZEOF_LONG==8
    return (long)pure[var[Nvar]] * hZeroMult(pn, sn, a, var, iv);
    #else
    int64 t=hZeroMult(pn, sn, a, var, iv);
    t *= pure[var[Nvar]];
    if ((t>=INT_MIN)&&(t<=INT_MAX)) sum=t;
    else if (!errorreported) WerrorS("int overflow in vdim 3");
    return sum;
    #endif
  }
  else
  {
    #if  SIZEOF_LONG==8
    sum = x * hZeroMult(pn, sn, a, var, iv);
    #else
    int64 t=hZeroMult(pn, sn, a, var, iv);
    t *= x;
    if ((t>=INT_MIN)&&(t<=INT_MAX)) sum=t;
    else if (!errorreported) WerrorS("int overflow in vdim 4");
    #endif
  }
  b = a;
  loop
  {
    a0 = a;
    x0 = x;
    hStepS(sn, Nstc, var, Nvar, &a, &x);
    hElimS(sn, &b, a0, a, var, iv);
    a1 = a;
    hPure(sn, a0, &a1, var, iv, pn, &i);
    hLex2S(sn, b, a0, a1, var, iv, hwork);
    b += (a1 - a0);
    if (a < Nstc)
    {
      #if  SIZEOF_LONG==8
      sum += (long)(x - x0) * hZeroMult(pn, sn, b, var, iv);
      #else
      int64 t=hZeroMult(pn, sn, b, var, iv);
      t *= (x-x0);
      t += sum;
      if ((t>=INT_MIN)&&(t<=INT_MAX)) sum=t;
      else if (!errorreported) WerrorS("int overflow in vdim 1");
      #endif
    }
    else
    {
      #if  SIZEOF_LONG==8
      sum += (long)(pure[var[Nvar]] - x0) * hZeroMult(pn, sn, b, var, iv);
      #else
      int64 t=hZeroMult(pn, sn, b, var, iv);
      t *= (pure[var[Nvar]]-x0);
      t += sum;
      if ((t>=INT_MIN)&&(t<=INT_MAX)) sum=t;
      else if (!errorreported) WerrorS("int overflow in vdim 2");
      #endif
      return sum;
    }
  }
}
 
static void hProject(scmon pure, varset sel)
{
  int  i, i0, k;
  i0 = 0;
  for (i = 1; i <= (currRing->N); i++)
  {
    if (pure[i])
    {
      i0++;
      sel[i0] = i;
    }
  }
  i = hNstc;
  memcpy(hwork, hstc, i * sizeof(scmon));
  hStaircase(hwork, &i, sel, i0);
  if ((i0 > 2) && (i > 10))
    hOrdSupp(hwork, i, sel, i0);
  memset(hpur0, 0, ((currRing->N) + 1) * sizeof(int));
  hPure(hwork, 0, &i, sel, i0, hpur0, &k);
  hLexS(hwork, i, sel, i0);
  hMu += hZeroMult(hpur0, hwork, i, sel, i0);
}
 
static void hDimMult(scmon pure, int Npure, scfmon rad, int Nrad,
 varset var, int Nvar)
{
  int  dn, iv, rad0, b, c, x;
  scmon pn;
  scfmon rn;
  if (Nrad < 2)
  {
    dn = Npure + Nrad;
    if (dn == hCo)
    {
      if (!Nrad)
        hProject(pure, hsel);
      else
      {
        pn = *rad;
        for (iv = Nvar; iv; iv--)
        {
          x = var[iv];
          if (pn[x])
          {
            pure[x] = 1;
            hProject(pure, hsel);
            pure[x] = 0;
          }
        }
      }
    }
    return;
  }
  iv = Nvar;
  dn = Npure+1;
  if (dn >= hCo)
  {
    if (dn > hCo)
      return;
    loop
    {
      if(!pure[var[iv]])
      {
        if(hNotZero(rad, Nrad, var, iv))
        {
          pure[var[iv]] = 1;
          hProject(pure, hsel);
          pure[var[iv]] = 0;
        }
      }
      iv--;
      if (!iv)
        return;
    }
  }
  while(pure[var[iv]]) iv--;
  hStepR(rad, Nrad, var, iv, &rad0);
  iv--;
  if (rad0 < Nrad)
  {
    pn = hGetpure(pure);
    rn = hGetmem(Nrad, rad, radmem[iv]);
    pn[var[iv + 1]] = 1;
    hDimMult(pn, Npure + 1, rn, rad0, var, iv);
    pn[var[iv + 1]] = 0;
    b = rad0;
    c = Nrad;
    hElimR(rn, &rad0, b, c, var, iv);
    hPure(rn, b, &c, var, iv, pn, &x);
    hLex2R(rn, rad0, b, c, var, iv, hwork);
    rad0 += (c - b);
    hDimMult(pn, Npure + x, rn, rad0, var, iv);
  }
  else
  {
    hDimMult(pure, Npure, rad, Nrad, var, iv);
  }
}
 
static void hDegree(ideal S, ideal Q)
{
  id_Test(S, currRing);
  if( Q!=NULL ) id_Test(Q, currRing);
 
  int  di;
  int  mc;
  hexist = hInit(S, Q, &hNexist);
  if (!hNexist)
  {
    hCo = 0;
    hMu = 1;
    return;
  }
  //hWeight();
  hwork = (scfmon)omAlloc(hNexist * sizeof(scmon));
  hvar = (varset)omAlloc(((currRing->N) + 1) * sizeof(int));
  hsel = (varset)omAlloc(((currRing->N) + 1) * sizeof(int));
  hpure = (scmon)omAlloc((1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  hpur0 = (scmon)omAlloc((1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  mc = hisModule;
  hrad = (scfmon)omAlloc(hNexist * sizeof(scmon));
  if (!mc)
  {
    memcpy(hrad, hexist, hNexist * sizeof(scmon));
    hstc = hexist;
    hNrad = hNstc = hNexist;
  }
  else
    hstc = (scfmon)omAlloc(hNexist * sizeof(scmon));
  radmem = hCreate((currRing->N) - 1);
  stcmem = hCreate((currRing->N) - 1);
  hCo = (currRing->N) + 1;
  di = hCo + 1;
  loop
  {
    if (mc)
    {
      hComp(hexist, hNexist, mc, hrad, &hNrad);
      hNstc = hNrad;
      memcpy(hstc, hrad, hNrad * sizeof(scmon));
    }
    if (hNrad)
    {
      hNvar = (currRing->N);
      hRadical(hrad, &hNrad, hNvar);
      hSupp(hrad, hNrad, hvar, &hNvar);
      if (hNvar)
      {
        hCo = hNvar;
        memset(hpure, 0, ((currRing->N) + 1) * sizeof(int));
        hPure(hrad, 0, &hNrad, hvar, hNvar, hpure, &hNpure);
        hLexR(hrad, hNrad, hvar, hNvar);
        hDimSolve(hpure, hNpure, hrad, hNrad, hvar, hNvar);
      }
    }
    else
    {
      hNvar = 1;
      hCo = 0;
    }
    if (hCo < di)
    {
      di = hCo;
      hMu = 0;
    }
    if (hNvar && (hCo == di))
    {
      if (di && (di < (currRing->N)))
        hDimMult(hpure, hNpure, hrad, hNrad, hvar, hNvar);
      else if (!di)
        hMu++;
      else
      {
        hStaircase(hstc, &hNstc, hvar, hNvar);
        if ((hNvar > 2) && (hNstc > 10))
          hOrdSupp(hstc, hNstc, hvar, hNvar);
        memset(hpur0, 0, ((currRing->N) + 1) * sizeof(int));
        hPure(hstc, 0, &hNstc, hvar, hNvar, hpur0, &hNpure);
        hLexS(hstc, hNstc, hvar, hNvar);
        hMu += hZeroMult(hpur0, hstc, hNstc, hvar, hNvar);
      }
    }
    mc--;
    if (mc <= 0)
      break;
  }
  hCo = di;
  hKill(stcmem, (currRing->N) - 1);
  hKill(radmem, (currRing->N) - 1);
  omFreeSize((ADDRESS)hpur0, (1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  omFreeSize((ADDRESS)hpure, (1 + ((currRing->N) * (currRing->N))) * sizeof(int));
  omFreeSize((ADDRESS)hsel, ((currRing->N) + 1) * sizeof(int));
  omFreeSize((ADDRESS)hvar, ((currRing->N) + 1) * sizeof(int));
  omFreeSize((ADDRESS)hwork, hNexist * sizeof(scmon));
  omFreeSize((ADDRESS)hrad, hNexist * sizeof(scmon));
  hDelete(hexist, hNexist);
  if (hisModule)
    omFreeSize((ADDRESS)hstc, hNexist * sizeof(scmon));
}
 
int  scMultInt(ideal S, ideal Q)
{
  id_Test(S, currRing);
  if( Q!=NULL ) id_Test(Q, currRing);
 
  hDegree(S, Q);
  return hMu;
}
 
void scPrintDegree(int co, int mu)
{
  int di = (currRing->N)-co;
  if (currRing->OrdSgn == 1)
  {
    if (di>0)
      Print("// dimension (proj.)  = %d\n// degree (proj.)   = %d\n", di-1, mu);
    else
      Print("// dimension (affine) = 0\n// degree (affine)  = %d\n",       mu);
  }
  else
    Print("// dimension (local)   = %d\n// multiplicity = %d\n", di, mu);
}
 
void scDegree(ideal S, intvec *modulweight, ideal Q)
{
  id_Test(S, currRing);
  if( Q!=NULL ) id_Test(Q, currRing);
 
  int co, mu, l;
  intvec *hseries2;
  intvec *hseries1 = hFirstSeries(S, modulweight, Q);
  if (errorreported) return;
  l = hseries1->length()-1;
  if (l > 1)
    hseries2 = hSecondSeries(hseries1);
  else
    hseries2 = hseries1;
  hDegreeSeries(hseries1, hseries2, &co, &mu);
  if ((l == 1) &&(mu == 0))
    scPrintDegree((currRing->N)+1, 0);
  else
    scPrintDegree(co, mu);
  if (l>1)
    delete hseries1;
  delete hseries2;
}
 
long  scMult0Int(ideal S, ideal Q)
{
  id_LmTest(S, currRing);
  if (Q!=NULL) id_LmTest(Q, currRing);
 
  int  mc;
  hexist = hInit(S, Q, &hNexist);
  if (!hNexist)
  {
    hMu = -1;
    return -1;
  }
  else
    hMu = 0;
 
  const ring r = currRing;
 
  hwork = (scfmon)omAlloc(hNexist * sizeof(scmon));
  hvar = (varset)omAlloc(((r->N) + 1) * sizeof(int));
  hpur0 = (scmon)omAlloc((1 + ((r->N) * (r->N))) * sizeof(int));
  mc = hisModule;
  if (!mc)
  {
    hstc = hexist;
    hNstc = hNexist;
  }
  else
    hstc = (scfmon)omAlloc(hNexist * sizeof(scmon));
  stcmem = hCreate((r->N) - 1);
  loop
  {
    if (mc)
    {
      hComp(hexist, hNexist, mc, hstc, &hNstc);
      if (!hNstc)
      {
        hMu = -1;
        break;
      }
    }
    hNvar = (r->N);
    for (int i = hNvar; i; i--)
      hvar[i] = i;
    hStaircase(hstc, &hNstc, hvar, hNvar);
    hSupp(hstc, hNstc, hvar, &hNvar);
    if ((hNvar == (r->N)) && (hNstc >= (r->N)))
    {
      if ((hNvar > 2) && (hNstc > 10))
        hOrdSupp(hstc, hNstc, hvar, hNvar);
      memset(hpur0, 0, ((r->N) + 1) * sizeof(int));
      hPure(hstc, 0, &hNstc, hvar, hNvar, hpur0, &hNpure);
      if (hNpure == hNvar)
      {
        hLexS(hstc, hNstc, hvar, hNvar);
        hMu += hZeroMult(hpur0, hstc, hNstc, hvar, hNvar);
      }
      else
        hMu = -1;
    }
    else if (hNvar)
      hMu = -1;
    mc--;
    if (mc <= 0 || hMu < 0)
      break;
  }
  hKill(stcmem, (r->N) - 1);
  omFreeSize((ADDRESS)hpur0, (1 + ((r->N) * (r->N))) * sizeof(int));
  omFreeSize((ADDRESS)hvar, ((r->N) + 1) * sizeof(int));
  omFreeSize((ADDRESS)hwork, hNexist * sizeof(scmon));
  hDelete(hexist, hNexist);
  if (hisModule)
    omFreeSize((ADDRESS)hstc, hNexist * sizeof(scmon));
  return hMu;
}
 
// HC
 
STATIC_VAR poly pWork;
 
static void hHedge(poly hEdge)
{
  pSetm(pWork);
  if (pLmCmp(pWork, hEdge) == currRing->OrdSgn)
  {
    for (int i = hNvar; i>0; i--)
      pSetExp(hEdge,i, pGetExp(pWork,i));
    pSetm(hEdge);
  }
}
 
 
static void hHedgeStep(scmon pure, scfmon stc,
                       int Nstc, varset var, int Nvar,poly hEdge)
{
  int  iv = Nvar -1, k = var[Nvar], a, a0, a1, b, i;
  int  x/*, x0*/;
  scmon pn;
  scfmon sn;
  if (iv==0)
  {
    pSetExp(pWork, k, pure[k]);
    hHedge(hEdge);
    return;
  }
  else if (Nstc==0)
  {
    for (i = Nvar; i>0; i--)
      pSetExp(pWork, var[i], pure[var[i]]);
    hHedge(hEdge);
    return;
  }
  x = a = 0;
  pn = hGetpure(pure);
  sn = hGetmem(Nstc, stc, stcmem[iv]);
  hStepS(sn, Nstc, var, Nvar, &a, &x);
  if (a == Nstc)
  {
    pSetExp(pWork, k, pure[k]);
    hHedgeStep(pn, sn, a, var, iv,hEdge);
    return;
  }
  else
  {
    pSetExp(pWork, k, x);
    hHedgeStep(pn, sn, a, var, iv,hEdge);
  }
  b = a;
  loop
  {
    a0 = a;
    // x0 = x;
    hStepS(sn, Nstc, var, Nvar, &a, &x);
    hElimS(sn, &b, a0, a, var, iv);
    a1 = a;
    hPure(sn, a0, &a1, var, iv, pn, &i);
    hLex2S(sn, b, a0, a1, var, iv, hwork);
    b += (a1 - a0);
    if (a < Nstc)
    {
      pSetExp(pWork, k, x);
      hHedgeStep(pn, sn, b, var, iv,hEdge);
    }
    else
    {
      pSetExp(pWork, k, pure[k]);
      hHedgeStep(pn, sn, b, var, iv,hEdge);
      return;
    }
  }
}
 
void scComputeHC(ideal S, ideal Q, int ak, poly &hEdge)
{
  id_LmTest(S, currRing);
  if (Q!=NULL) id_LmTest(Q, currRing);
 
  int  i;
  int  k = ak;
  #ifdef HAVE_RINGS
  if (rField_is_Ring(currRing) && (currRing->OrdSgn == -1))
  {
    //consider just monic generators (over rings with zero-divisors)
    ideal SS=id_Head(S,currRing);
    for(i=0;i<=idElem(S);i++)
    {
      if((SS->m[i]!=NULL)
      && ((p_IsPurePower(SS->m[i],currRing)==0)
        ||(!n_IsUnit(pGetCoeff(SS->m[i]), currRing->cf))))
      {
        p_Delete(&SS->m[i],currRing);
      }
    }
    S=id_Copy(SS,currRing);
    idSkipZeroes(S);
  }
  #if 0
  printf("\nThis is HC:\n");
  for(int ii=0;ii<=idElem(S);ii++)
  {
    pWrite(S->m[ii]);
  }
  //getchar();
  #endif
  #endif
  if(idElem(S) == 0)
    return;
  hNvar = (currRing->N);
  hexist = hInit(S, Q, &hNexist);
  if (k!=0)
    hComp(hexist, hNexist, k, hexist, &hNstc);
  else
    hNstc = hNexist;
  assume(hNexist > 0);
  hwork = (scfmon)omAlloc(hNexist * sizeof(scmon));
  hvar = (varset)omAlloc((hNvar + 1) * sizeof(int));
  hpure = (scmon)omAlloc((1 + (hNvar * hNvar)) * sizeof(int));
  stcmem = hCreate(hNvar - 1);
  for (i = hNvar; i>0; i--)
    hvar[i] = i;
  hStaircase(hexist, &hNstc, hvar, hNvar);
  if ((hNvar > 2) && (hNstc > 10))
    hOrdSupp(hexist, hNstc, hvar, hNvar);
  memset(hpure, 0, (hNvar + 1) * sizeof(int));
  hPure(hexist, 0, &hNstc, hvar, hNvar, hpure, &hNpure);
  hLexS(hexist, hNstc, hvar, hNvar);
  if (hEdge!=NULL)
    pLmFree(hEdge);
  hEdge = pInit();
  pWork = pInit();
  hHedgeStep(hpure, hexist, hNstc, hvar, hNvar,hEdge);
  pSetComp(hEdge,ak);
  hKill(stcmem, hNvar - 1);
  omFreeSize((ADDRESS)hwork, hNexist * sizeof(scmon));
  omFreeSize((ADDRESS)hvar, (hNvar + 1) * sizeof(int));
  omFreeSize((ADDRESS)hpure, (1 + (hNvar * hNvar)) * sizeof(int));
  hDelete(hexist, hNexist);
  pLmFree(pWork);
}
 
 
 
//  kbase
 
STATIC_VAR poly last;
STATIC_VAR scmon act;
 
static void scElKbase()
{
  poly q = pInit();
  pSetCoeff0(q,nInit(1));
  pSetExpV(q,act);
  pNext(q) = NULL;
  last = pNext(last) = q;
}
 
static int scMax( int i, scfmon stc, int Nvar)
{
  int x, y=stc[0][Nvar];
  for (; i;)
  {
    i--;
    x = stc[i][Nvar];
    if (x > y) y = x;
  }
  return y;
}
 
static int scMin( int i, scfmon stc, int Nvar)
{
  int x, y=stc[0][Nvar];
  for (; i;)
  {
    i--;
    x = stc[i][Nvar];
    if (x < y) y = x;
  }
  return y;
}
 
static int scRestrict( int &Nstc, scfmon stc, int Nvar)
{
  int x, y;
  int i, j, Istc = Nstc;
 
  y = MAX_INT_VAL;
  for (i=Nstc-1; i>=0; i--)
  {
    j = Nvar-1;
    loop
    {
      if(stc[i][j] != 0) break;
      j--;
      if (j == 0)
      {
        Istc--;
        x = stc[i][Nvar];
        if (x < y) y = x;
        stc[i] = NULL;
        break;
      }
    }
  }
  if (Istc < Nstc)
  {
    for (i=Nstc-1; i>=0; i--)
    {
      if (stc[i] && (stc[i][Nvar] >= y))
      {
        Istc--;
        stc[i] = NULL;
      }
    }
    j = 0;
    while (stc[j]) j++;
    i = j+1;
    for(; i<Nstc; i++)
    {
      if (stc[i])
      {
        stc[j] = stc[i];
        j++;
      }
    }
    Nstc = Istc;
    return y;
  }
  else
    return -1;
}
 
static void scAll( int Nvar, int deg)
{
  int i;
  int d = deg;
  if (d == 0)
  {
    for (i=Nvar; i; i--) act[i] = 0;
    scElKbase();
    return;
  }
  if (Nvar == 1)
  {
    act[1] = d;
    scElKbase();
    return;
  }
  do
  {
    act[Nvar] = d;
    scAll(Nvar-1, deg-d);
    d--;
  } while (d >= 0);
}
 
static void scAllKbase( int Nvar, int ideg, int deg)
{
  do
  {
    act[Nvar] = ideg;
    scAll(Nvar-1, deg-ideg);
    ideg--;
  } while (ideg >= 0);
}
 
static void scDegKbase( scfmon stc, int Nstc, int Nvar, int deg)
{
  int  Ivar, Istc, i, j;
  scfmon sn;
  int x, ideg;
 
  if (deg == 0)
  {
    for (i=Nstc-1; i>=0; i--)
    {
      for (j=Nvar;j;j--){ if(stc[i][j]) break; }
      if (j==0){return;}
    }
    for (i=Nvar; i; i--) act[i] = 0;
    scElKbase();
    return;
  }
  if (Nvar == 1)
  {
    for (i=Nstc-1; i>=0; i--) if(deg >= stc[i][1]) return;
    act[1] = deg;
    scElKbase();
    return;
  }
  Ivar = Nvar-1;
  sn = hGetmem(Nstc, stc, stcmem[Ivar]);
  x = scRestrict(Nstc, sn, Nvar);
  if (x <= 0)
  {
    if (x == 0) return;
    ideg = deg;
  }
  else
  {
    if (deg < x) ideg = deg;
    else ideg = x-1;
    if (Nstc == 0)
    {
      scAllKbase(Nvar, ideg, deg);
      return;
    }
  }
  loop
  {
    x = scMax(Nstc, sn, Nvar);
    while (ideg >= x)
    {
      act[Nvar] = ideg;
      scDegKbase(sn, Nstc, Ivar, deg-ideg);
      ideg--;
    }
    if (ideg < 0) return;
    Istc = Nstc;
    for (i=Nstc-1; i>=0; i--)
    {
      if (ideg < sn[i][Nvar])
      {
        Istc--;
        sn[i] = NULL;
      }
    }
    if (Istc == 0)
    {
      scAllKbase(Nvar, ideg, deg);
      return;
    }
    j = 0;
    while (sn[j]) j++;
    i = j+1;
    for (; i<Nstc; i++)
    {
      if (sn[i])
      {
        sn[j] = sn[i];
        j++;
      }
    }
    Nstc = Istc;
  }
}
 
static void scInKbase( scfmon stc, int Nstc, int Nvar)
{
  int  Ivar, Istc, i, j;
  scfmon sn;
  int x, ideg;
 
  if (Nvar == 1)
  {
    ideg = scMin(Nstc, stc, 1);
    while (ideg > 0)
    {
      ideg--;
      act[1] = ideg;
      scElKbase();
    }
    return;
  }
  Ivar = Nvar-1;
  sn = hGetmem(Nstc, stc, stcmem[Ivar]);
  x = scRestrict(Nstc, sn, Nvar);
  if (x == 0) return;
  ideg = x-1;
  loop
  {
    x = scMax(Nstc, sn, Nvar);
    while (ideg >= x)
    {
      act[Nvar] = ideg;
      scInKbase(sn, Nstc, Ivar);
      ideg--;
    }
    if (ideg < 0) return;
    Istc = Nstc;
    for (i=Nstc-1; i>=0; i--)
    {
      if (ideg < sn[i][Nvar])
      {
        Istc--;
        sn[i] = NULL;
      }
    }
    j = 0;
    while (sn[j]) j++;
    i = j+1;
    for (; i<Nstc; i++)
    {
      if (sn[i])
      {
        sn[j] = sn[i];
        j++;
      }
    }
    Nstc = Istc;
  }
}
 
static ideal scIdKbase(poly q, const int rank)
{
  ideal res = idInit(pLength(q), rank);
  polyset mm = res->m;
  do
  {
    *mm = q; ++mm;
 
    const poly p = pNext(q);
    pNext(q) = NULL;
    q = p;
 
  } while (q!=NULL);
 
  id_Test(res, currRing);   // WRONG RANK!!!???
  return res;
}
 
ideal scKBase(int deg, ideal s, ideal Q, intvec * mv)
{
  if( Q!=NULL) id_Test(Q, currRing);
 
  int  i, di;
  poly p;
 
  if (deg < 0)
  {
    di = scDimInt(s, Q);
    if (di != 0)
    {
      //Werror("KBase not finite");
      return idInit(1,s->rank);
    }
  }
  stcmem = hCreate((currRing->N) - 1);
  hexist = hInit(s, Q, &hNexist);
  p = last = pInit();
  /*pNext(p) = NULL;*/
  act = (scmon)omAlloc(((currRing->N) + 1) * sizeof(int));
  *act = 0;
  if (!hNexist)
  {
    scAll((currRing->N), deg);
    goto ende;
  }
  if (!hisModule)
  {
    if (deg < 0) scInKbase(hexist, hNexist, (currRing->N));
    else scDegKbase(hexist, hNexist, (currRing->N), deg);
  }
  else
  {
    hstc = (scfmon)omAlloc(hNexist * sizeof(scmon));
    for (i = 1; i <= hisModule; i++)
    {
      *act = i;
      hComp(hexist, hNexist, i, hstc, &hNstc);
      int deg_ei=deg;
      if (mv!=NULL) deg_ei -= (*mv)[i-1];
      if ((deg < 0) || (deg_ei>=0))
      {
        if (hNstc)
        {
          if (deg < 0) scInKbase(hstc, hNstc, (currRing->N));
          else scDegKbase(hstc, hNstc, (currRing->N), deg_ei);
        }
        else
          scAll((currRing->N), deg_ei);
      }
    }
    omFreeSize((ADDRESS)hstc, hNexist * sizeof(scmon));
  }
ende:
  hDelete(hexist, hNexist);
  omFreeSize((ADDRESS)act, ((currRing->N) + 1) * sizeof(int));
  hKill(stcmem, (currRing->N) - 1);
  pLmFree(&p);
  if (p == NULL)
    return idInit(1,s->rank);
 
  last = p;
  return scIdKbase(p, s->rank);
}
 
#if 0 //-- alternative implementation of scComputeHC
/*
void scComputeHCw(ideal ss, ideal Q, int ak, poly &hEdge)
{
  id_LmTest(ss, currRing);
  if (Q!=NULL) id_LmTest(Q, currRing);
 
  int  i, di;
  poly p;
 
  if (hEdge!=NULL)
    pLmFree(hEdge);
 
  ideal s=idInit(IDELEMS(ss),ak);
  for(i=IDELEMS(ss)-1;i>=0;i--)
  {
    if (ss->m[i]!=NULL) s->m[i]=pHead(ss->m[i]);
  }
  di = scDimInt(s, Q);
  stcmem = hCreate((currRing->N) - 1);
  hexist = hInit(s, Q, &hNexist);
  p = last = pInit();
  // pNext(p) = NULL;
  act = (scmon)omAlloc(((currRing->N) + 1) * sizeof(int));
  *act = 0;
  if (!hNexist)
  {
    scAll((currRing->N), -1);
    goto ende;
  }
  if (!hisModule)
  {
    scInKbase(hexist, hNexist, (currRing->N));
  }
  else
  {
    hstc = (scfmon)omAlloc(hNexist * sizeof(scmon));
    for (i = 1; i <= hisModule; i++)
    {
      *act = i;
      hComp(hexist, hNexist, i, hstc, &hNstc);
      if (hNstc)
      {
        scInKbase(hstc, hNstc, (currRing->N));
      }
      else
        scAll((currRing->N), -1);
    }
    omFreeSize((ADDRESS)hstc, hNexist * sizeof(scmon));
  }
ende:
  hDelete(hexist, hNexist);
  omFreeSize((ADDRESS)act, ((currRing->N) + 1) * sizeof(int));
  hKill(stcmem, (currRing->N) - 1);
  pDeleteLm(&p);
  idDelete(&s);
  if (p == NULL)
  {
    return; // no HEdge
  }
  else
  {
    last = p;
    ideal res=scIdKbase(p, ss->rank);
    poly p_ind=res->m[0]; int ind=0;
    for(i=IDELEMS(res)-1;i>0;i--)
    {
      if (pCmp(res->m[i],p_ind)==-1) { p_ind=res->m[i]; ind=i; }
    }
    assume(p_ind!=NULL);
    assume(res->m[ind]==p_ind);
    hEdge=p_ind;
    res->m[ind]=NULL;
    nDelete(&pGetCoeff(hEdge));
    pGetCoeff(hEdge)=NULL;
    for(i=(currRing->N);i>0;i--)
      pIncrExp(hEdge,i);
    pSetm(hEdge);
 
    idDelete(&res);
    return;
  }
}
 */
#endif
 
#ifdef HAVE_SHIFTBBA
 
/*
 * Computation of the Gel'fand-Kirillov Dimension
 */
 
#include "polys/shiftop.h"
#include <vector>
 
static std::vector<int> countCycles(const intvec* _G, int v, std::vector<int> path, std::vector<BOOLEAN> visited, std::vector<BOOLEAN> cyclic, std::vector<int> cache)
{
  intvec* G = ivCopy(_G); // modifications must be local
 
  if (cache[v] != -2) return cache; // value is already cached
 
  visited[v] = TRUE;
  path.push_back(v);
 
  int cycles = 0;
  for (int w = 0; w < G->cols(); w++)
  {
    if (IMATELEM(*G, v + 1, w + 1)) // edge v -> w exists in G
    {
      if (!visited[w])
      { // continue with w
        cache = countCycles(G, w, path, visited, cyclic, cache);
        if (cache[w] == -1)
        {
          cache[v] = -1;
          return cache;
        }
        cycles = si_max(cycles, cache[w]);
      }
      else
      { // found new cycle
        int pathIndexOfW = -1;
        for (int i = path.size() - 1; i >= 0; i--) {
          if (cyclic[path[i]] == 1) { // found an already cyclic vertex
            cache[v] = -1;
            return cache;
          }
          cyclic[path[i]] = TRUE;
 
          if (path[i] == w) { // end of the cycle
            assume(IMATELEM(*G, v + 1, w + 1) != 0);
            IMATELEM(*G, v + 1, w + 1) = 0; // remove edge v -> w
            pathIndexOfW = i;
            break;
          } else {
            assume(IMATELEM(*G, path[i - 1] + 1, path[i] + 1) != 0);
            IMATELEM(*G, path[i - 1] + 1, path[i] + 1) = 0; // remove edge vi-1 -> vi
          }
        }
        assume(pathIndexOfW != -1); // should never happen
        for (int i = path.size() - 1; i >= pathIndexOfW; i--) {
          cache = countCycles(G, path[i], path, visited, cyclic, cache);
          if (cache[path[i]] == -1)
          {
            cache[v] = -1;
            return cache;
          }
          cycles = si_max(cycles, cache[path[i]] + 1);
        }
      }
    }
  }
  cache[v] = cycles;
 
  delete G;
  return cache;
}
 
// -1 is infinity
static int graphGrowth(const intvec* G)
{
  // init
  int n = G->cols();
  std::vector<int> path;
  std::vector<BOOLEAN> visited;
  std::vector<BOOLEAN> cyclic;
  std::vector<int> cache;
  visited.resize(n, FALSE);
  cyclic.resize(n, FALSE);
  cache.resize(n, -2);
 
  // get max number of cycles
  int cycles = 0;
  for (int v = 0; v < n; v++)
  {
    cache = countCycles(G, v, path, visited, cyclic, cache);
    if (cache[v] == -1)
      return -1;
    cycles = si_max(cycles, cache[v]);
  }
  return cycles;
}
 
// ATTENTION:
//  - `words` contains the words normal modulo M of length n
//  - `numberOfNormalWords` contains the number of words normal modulo M of length 0 ... n
static void _lp_computeNormalWords(ideal words, int& numberOfNormalWords, int length, ideal M, int minDeg, int& last)
{
  if (length <= 0){
    poly one = pOne();
    if (p_LPDivisibleBy(M, one, currRing)) // 1 \in M => no normal words at all
    {
      pDelete(&one);
      last = -1;
      numberOfNormalWords = 0;
    }
    else
    {
      words->m[0] = one;
      last = 0;
      numberOfNormalWords = 1;
    }
    return;
  }
 
  _lp_computeNormalWords(words, numberOfNormalWords, length - 1, M, minDeg, last);
 
  int nVars = currRing->isLPring - currRing->LPncGenCount;
  int numberOfNewNormalWords = 0;
 
  for (int j = nVars - 1; j >= 0; j--)
  {
    for (int i = last; i >= 0; i--)
    {
      int index = (j * (last + 1)) + i;
 
      if (words->m[i] != NULL)
      {
        if (j > 0) {
          words->m[index] = pCopy(words->m[i]);
        }
 
        int varOffset = ((length - 1) * currRing->isLPring) + 1;
        pSetExp(words->m[index], varOffset + j, 1);
        pSetm(words->m[index]);
        pTest(words->m[index]);
 
        if (length >= minDeg && p_LPDivisibleBy(M, words->m[index], currRing))
        {
          pDelete(&words->m[index]);
          words->m[index] = NULL;
        }
        else
        {
          numberOfNewNormalWords++;
        }
      }
    }
  }
 
  last = nVars * last + nVars - 1;
 
  numberOfNormalWords += numberOfNewNormalWords;
}
 
static ideal lp_computeNormalWords(int length, ideal M)
{
  long minDeg = IDELEMS(M) > 0 ? pTotaldegree(M->m[0]) : 0;
  for (int i = 1; i < IDELEMS(M); i++)
  {
    minDeg = si_min(minDeg, pTotaldegree(M->m[i]));
  }
 
  int nVars = currRing->isLPring - currRing->LPncGenCount;
 
  int maxElems = 1;
  for (int i = 0; i < length; i++) // maxElems = nVars^n
    maxElems *= nVars;
  ideal words = idInit(maxElems);
  int last, numberOfNormalWords;
  _lp_computeNormalWords(words, numberOfNormalWords, length, M, minDeg, last);
  idSkipZeroes(words);
  return words;
}
 
static int lp_countNormalWords(int upToLength, ideal M)
{
  long minDeg = IDELEMS(M) > 0 ? pTotaldegree(M->m[0]) : 0;
  for (int i = 1; i < IDELEMS(M); i++)
  {
    minDeg = si_min(minDeg, pTotaldegree(M->m[i]));
  }
 
  int nVars = currRing->isLPring - currRing->LPncGenCount;
 
  int maxElems = 1;
  for (int i = 0; i < upToLength; i++) // maxElems = nVars^n
    maxElems *= nVars;
  ideal words = idInit(maxElems);
  int last, numberOfNormalWords;
  _lp_computeNormalWords(words, numberOfNormalWords, upToLength, M, minDeg, last);
  idDelete(&words);
  return numberOfNormalWords;
}
 
// NULL if graph is undefined
intvec* lp_ufnarovskiGraph(ideal G, ideal &standardWords)
{
  long l = 0;
  for (int i = 0; i < IDELEMS(G); i++)
    l = si_max(pTotaldegree(G->m[i]), l);
  l--;
  if (l <= 0)
  {
    WerrorS("Ufnarovski graph not implemented for l <= 0");
    return NULL;
  }
  int lV = currRing->isLPring;
 
  standardWords = lp_computeNormalWords(l, G);
 
  int n = IDELEMS(standardWords);
  intvec* UG = new intvec(n, n, 0);
  for (int i = 0; i < n; i++)
  {
    for (int j = 0; j < n; j++)
    {
      poly v = standardWords->m[i];
      poly w = standardWords->m[j];
 
      // check whether v*x1 = x2*w (overlap)
      bool overlap = true;
      for (int k = 1; k <= (l - 1) * lV; k++)
      {
        if (pGetExp(v, k + lV) != pGetExp(w, k)) {
          overlap = false;
          break;
        }
      }
 
      if (overlap)
      {
        // create the overlap
        poly p = pMult(pCopy(v), p_LPVarAt(w, l, currRing));
 
        // check whether the overlap is normal
        bool normal = true;
        for (int k = 0; k < IDELEMS(G); k++)
        {
          if (p_LPDivisibleBy(G->m[k], p, currRing))
          {
            normal = false;
            break;
          }
        }
 
        if (normal)
        {
          IMATELEM(*UG, i + 1, j + 1) = 1;
        }
      }
    }
  }
  return UG;
}
 
// -1 is infinity, -2 is error
int lp_gkDim(const ideal _G)
{
  id_Test(_G, currRing);
 
  if (rField_is_Ring(currRing)) {
      WerrorS("GK-Dim not implemented for rings");
      return -2;
  }
 
  for (int i=IDELEMS(_G)-1;i>=0; i--)
  {
    if (_G->m[i] != NULL)
    {
      if (pGetComp(_G->m[i]) != 0)
      {
        WerrorS("GK-Dim not implemented for modules");
        return -2;
      }
      if (pGetNCGen(_G->m[i]) != 0)
      {
        WerrorS("GK-Dim not implemented for bi-modules");
        return -2;
      }
    }
  }
 
  ideal G = id_Head(_G, currRing); // G = LM(G) (and copy)
  idSkipZeroes(G); // remove zeros
  id_DelLmEquals(G, currRing); // remove duplicates
 
  // check if G is the zero ideal
  if (IDELEMS(G) == 1 && G->m[0] == NULL)
  {
    // NOTE: this is needed because if the ideal is <0>, then idSkipZeroes keeps this element, and IDELEMS is still 1!
    int lV = currRing->isLPring;
    int ncGenCount = currRing->LPncGenCount;
    if (lV - ncGenCount == 0)
    {
      idDelete(&G);
      return 0;
    }
    if (lV - ncGenCount == 1)
    {
      idDelete(&G);
      return 1;
    }
    if (lV - ncGenCount >= 2)
    {
      idDelete(&G);
      return -1;
    }
  }
 
  // get the max deg
  long maxDeg = 0;
  for (int i = 0; i < IDELEMS(G); i++)
  {
    maxDeg = si_max(maxDeg, pTotaldegree(G->m[i]));
 
    // also check whether G = <1>
    if (pIsConstantComp(G->m[i]))
    {
      WerrorS("GK-Dim not defined for 0-ring");
      idDelete(&G);
      return -2;
    }
  }
 
  // early termination if G \subset X
  if (maxDeg <= 1)
  {
    int lV = currRing->isLPring;
    int ncGenCount = currRing->LPncGenCount;
    if (IDELEMS(G) == lV - ncGenCount) // V = {1} no edges
    {
      idDelete(&G);
      return 0;
    }
    if (IDELEMS(G) == lV - ncGenCount - 1) // V = {1} with loop
    {
      idDelete(&G);
      return 1;
    }
    if (IDELEMS(G) <= lV - ncGenCount - 2) // V = {1} with more than one loop
    {
      idDelete(&G);
      return -1;
    }
  }
 
  ideal standardWords;
  intvec* UG = lp_ufnarovskiGraph(G, standardWords);
  if (UG == NULL)
  {
    idDelete(&G);
    return -2;
  }
  if (errorreported)
  {
    delete UG;
    idDelete(&G);
    return -2;
  }
  int gkDim = graphGrowth(UG);
  delete UG;
  idDelete(&G);
  return gkDim;
}
 
// converts an intvec matrix to a vector<vector<int> >
static std::vector<std::vector<int> > iv2vv(intvec* M)
{
  int rows = M->rows();
  int cols = M->cols();
 
  std::vector<std::vector<int> > mat(rows, std::vector<int>(cols));
 
  for (int i = 0; i < rows; i++)
  {
    for (int j = 0; j < cols; j++)
    {
      mat[i][j] = IMATELEM(*M, i + 1, j + 1);
    }
  }
 
  return mat;
}
 
static void vvPrint(const std::vector<std::vector<int> >& mat)
{
  for (int i = 0; i < mat.size(); i++)
  {
    for (int j = 0; j < mat[i].size(); j++)
    {
      Print("%d ", mat[i][j]);
    }
    PrintLn();
  }
}
 
static void vvTest(const std::vector<std::vector<int> >& mat)
{
  if (mat.size() > 0)
  {
    int cols = mat[0].size();
    for (int i = 1; i < mat.size(); i++)
    {
      if (cols != mat[i].size())
        WerrorS("number of cols in matrix inconsistent");
    }
  }
}
 
static void vvDeleteRow(std::vector<std::vector<int> >& mat, int row)
{
  mat.erase(mat.begin() + row);
}
 
static void vvDeleteColumn(std::vector<std::vector<int> >& mat, int col)
{
  for (int i = 0; i < mat.size(); i++)
  {
    mat[i].erase(mat[i].begin() + col);
  }
}
 
static BOOLEAN vvIsRowZero(const std::vector<std::vector<int> >& mat, int row)
{
  for (int i = 0; i < mat[row].size(); i++)
  {
    if (mat[row][i] != 0)
      return FALSE;
  }
  return TRUE;
}
 
static BOOLEAN vvIsColumnZero(const std::vector<std::vector<int> >& mat, int col)
{
  for (int i = 0; i < mat.size(); i++)
  {
    if (mat[i][col] != 0)
      return FALSE;
  }
  return TRUE;
}
 
static BOOLEAN vvIsZero(const std::vector<std::vector<int> >& mat)
{
  for (int i = 0; i < mat.size(); i++)
  {
    if (!vvIsRowZero(mat, i))
      return FALSE;
  }
  return TRUE;
}
 
static std::vector<std::vector<int> > vvMult(const std::vector<std::vector<int> >& a, const std::vector<std::vector<int> >& b)
{
  int ra = a.size();
  int rb = b.size();
  int ca = a.size() > 0 ? a[0].size() : 0;
  int cb = b.size() > 0 ? b[0].size() : 0;
 
  if (ca != rb)
  {
    WerrorS("matrix dimensions do not match");
    return std::vector<std::vector<int> >();
  }
 
  std::vector<std::vector<int> > res(ra, std::vector<int>(cb));
  for (int i = 0; i < ra; i++)
  {
    for (int j = 0; j < cb; j++)
    {
      int sum = 0;
      for (int k = 0; k < ca; k++)
        sum += a[i][k] * b[k][j];
      res[i][j] = sum;
    }
  }
  return res;
}
 
static BOOLEAN isAcyclic(const intvec* G)
{
  // init
  int n = G->cols();
  std::vector<int> path;
  std::vector<BOOLEAN> visited;
  std::vector<BOOLEAN> cyclic;
  std::vector<int> cache;
  visited.resize(n, FALSE);
  cyclic.resize(n, FALSE);
  cache.resize(n, -2);
 
  for (int v = 0; v < n; v++)
  {
    cache = countCycles(G, v, path, visited, cyclic, cache);
    // check that there are 0 cycles from v
    if (cache[v] != 0)
      return FALSE;
  }
  return TRUE;
}
 
/*
 * Computation of the K-Dimension
 */
 
// -1 is infinity, -2 is error
int lp_kDim(const ideal _G)
{
  if (rField_is_Ring(currRing)) {
      WerrorS("K-Dim not implemented for rings");
      return -2;
  }
 
  for (int i=IDELEMS(_G)-1;i>=0; i--)
  {
    if (_G->m[i] != NULL)
    {
      if (pGetComp(_G->m[i]) != 0)
      {
        WerrorS("K-Dim not implemented for modules");
        return -2;
      }
      if (pGetNCGen(_G->m[i]) != 0)
      {
        WerrorS("K-Dim not implemented for bi-modules");
        return -2;
      }
    }
  }
 
  ideal G = id_Head(_G, currRing); // G = LM(G) (and copy)
  if (TEST_OPT_PROT)
    Print("%d original generators\n", IDELEMS(G));
  idSkipZeroes(G); // remove zeros
  id_DelLmEquals(G, currRing); // remove duplicates
  if (TEST_OPT_PROT)
    Print("%d non-zero unique generators\n", IDELEMS(G));
 
  // check if G is the zero ideal
  if (IDELEMS(G) == 1 && G->m[0] == NULL)
  {
    // NOTE: this is needed because if the ideal is <0>, then idSkipZeroes keeps this element, and IDELEMS is still 1!
    int lV = currRing->isLPring;
    int ncGenCount = currRing->LPncGenCount;
    if (lV - ncGenCount == 0)
    {
      idDelete(&G);
      return 1;
    }
    if (lV - ncGenCount == 1)
    {
      idDelete(&G);
      return -1;
    }
    if (lV - ncGenCount >= 2)
    {
      idDelete(&G);
      return -1;
    }
  }
 
  // get the max deg
  long maxDeg = 0;
  for (int i = 0; i < IDELEMS(G); i++)
  {
    maxDeg = si_max(maxDeg, pTotaldegree(G->m[i]));
 
    // also check whether G = <1>
    if (pIsConstantComp(G->m[i]))
    {
      WerrorS("K-Dim not defined for 0-ring"); // TODO is it minus infinity ?
      idDelete(&G);
      return -2;
    }
  }
  if (TEST_OPT_PROT)
    Print("max deg: %ld\n", maxDeg);
 
 
  // for normal words of length minDeg ... maxDeg-1
  // brute-force the normal words
  if (TEST_OPT_PROT)
    PrintS("Computing normal words normally...\n");
  long numberOfNormalWords = lp_countNormalWords(maxDeg - 1, G);
 
  if (TEST_OPT_PROT)
    Print("%ld normal words up to length %ld\n", numberOfNormalWords, maxDeg - 1);
 
  // early termination if G \subset X
  if (maxDeg <= 1)
  {
    int lV = currRing->isLPring;
    int ncGenCount = currRing->LPncGenCount;
    if (IDELEMS(G) == lV - ncGenCount) // V = {1} no edges
    {
      idDelete(&G);
      return numberOfNormalWords;
    }
    if (IDELEMS(G) == lV - ncGenCount - 1) // V = {1} with loop
    {
      idDelete(&G);
      return -1;
    }
    if (IDELEMS(G) <= lV - ncGenCount - 2) // V = {1} with more than one loop
    {
      idDelete(&G);
      return -1;
    }
  }
 
  if (TEST_OPT_PROT)
    PrintS("Computing Ufnarovski graph...\n");
 
  ideal standardWords;
  intvec* UG = lp_ufnarovskiGraph(G, standardWords);
  if (UG == NULL)
  {
    idDelete(&G);
    return -2;
  }
  if (errorreported)
  {
    delete UG;
    idDelete(&G);
    return -2;
  }
 
  if (TEST_OPT_PROT)
    Print("Ufnarovski graph is %dx%d.\n", UG->rows(), UG->cols());
 
  if (TEST_OPT_PROT)
    PrintS("Checking whether Ufnarovski graph is acyclic...\n");
 
  if (!isAcyclic(UG))
  {
    // in this case we have infinitely many normal words
    return -1;
  }
 
  std::vector<std::vector<int> > vvUG = iv2vv(UG);
  for (int i = 0; i < vvUG.size(); i++)
  {
    if (vvIsRowZero(vvUG, i) && vvIsColumnZero(vvUG, i)) // i is isolated vertex
    {
      vvDeleteRow(vvUG, i);
      vvDeleteColumn(vvUG, i);
      i--;
    }
  }
  if (TEST_OPT_PROT)
    Print("Simplified Ufnarovski graph to %dx%d.\n", (int)vvUG.size(), (int)vvUG.size());
 
  // for normal words of length >= maxDeg
  // use Ufnarovski graph
  if (TEST_OPT_PROT)
    PrintS("Computing normal words via Ufnarovski graph...\n");
  std::vector<std::vector<int> > UGpower = vvUG;
  long nUGpower = 1;
  while (!vvIsZero(UGpower))
  {
    if (TEST_OPT_PROT)
      PrintS("Start count graph entries.\n");
    for (int i = 0; i < UGpower.size(); i++)
    {
      for (int j = 0; j < UGpower[i].size(); j++)
      {
        numberOfNormalWords += UGpower[i][j];
      }
    }
 
    if (TEST_OPT_PROT)
    {
      PrintS("Done count graph entries.\n");
      Print("%ld normal words up to length %ld\n", numberOfNormalWords, maxDeg - 1 + nUGpower);
    }
 
    if (TEST_OPT_PROT)
      PrintS("Start mat mult.\n");
    UGpower = vvMult(UGpower, vvUG); // TODO: avoid creation of new intvec
    if (TEST_OPT_PROT)
      PrintS("Done mat mult.\n");
    nUGpower++;
  }
 
  delete UG;
  idDelete(&G);
  return numberOfNormalWords;
}
#endif