source: git/Singular/LIB/reszeta.lib @ 1e1ec4

/////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Algebraic Geometry";
info="
LIBRARY:  reszeta.lib       topological Zeta-function and
                            some other applications of desingularization

AUTHORS:  A. Fruehbis-Krueger,  anne@mathematik.uni-kl.de,
@*        G. Pfister,           pfister@mathematik.uni-kl.de

REFERENCES:
[1] Fruehbis-Krueger,A., Pfister,G.: Some Applications of Resolution of
@*  Singularities from a Practical Point of View, in Computational
@*  Commutative and Non-commutative Algebraic Geometry,
@*  NATO Science Series III, Computer and Systems Sciences 196, 104-117 (2005)
[2] Fruehbis-Krueger: An Application of Resolution of Singularities:
@*  Computing the topological Zeta-function of isolated surface singularities
@*  in (C^3,0),  in D.Cheniot, N.Dutertre et al.(Editors): Singularity Theory, @*  World Scientific Publishing (2007)

PROCEDURES:
intersectionDiv(L)   computes intersection form and genera of exceptional
                     divisors (isolated singularities of surfaces)
spectralNeg(L)       computes negative spectral numbers
                     (isolated hypersurface singularity)
discrepancy(L)       computes discrepancy of given resolution
zetaDL(L,d)          computes Denef-Loeser zeta function
                     (hypersurface singularity of dimension 2)
collectDiv(L[,iv])   identify exceptional divisors in different charts
                     (embedded and non-embedded case)
prepEmbDiv(L[,b])    prepare list of divisors (including components
                     of strict transform, embedded case)
abstractR(L)         pass from embedded to non-embedded resolution
computeV(re,DL)      multiplicities of divisors in pullback of volume form
computeN(re,DL)      multiplicities of divisors in total transform of resolution
";
LIB "resolve.lib";
LIB "solve.lib";
LIB "normal.lib";
///////////////////////////////////////////////////////////////////////////////
static proc spectral1(poly h,list re, list DL,intvec v, intvec n)
"Internal procedure - no help and no example available
"
{
//--- compute one spectral number
//--- DL is output of prepEmbDiv
   int i;
   intvec w=computeH(h,re,DL);
   number gw=number(w[1]+v[1])/number(n[1]);
   for(i=2;i<=size(v);i++)
   {
      if(gw>number(w[i]+v[i])/number(n[i]))
      {
         gw=number(w[i]+v[i])/number(n[i]);
      }
   }
   return(gw-1);
}

///////////////////////////////////////////////////////////////////////////////

proc spectralNeg(list re,list #)
"USAGE:   spectralNeg(L);
@*        L = list of rings
ASSUME:   L is output of resolution of singularities
RETURN:   list of numbers, each a spectral number in (-1,0]
EXAMPLE:  example spectralNeg;  shows an example
"
{
//-----------------------------------------------------------------------------
// Initialization and Sanity Checks
//-----------------------------------------------------------------------------
   int i,j,l;
   number bound;
   list resu;
   if(size(#)>0)
   {
//--- undocumented feature:
//--- if # is not empty it computes numbers up to this bound,
//--- not necessarily spectral numbers
      bound=number(#[1]);
   }
//--- get list of embedded divisors
   list DL=prepEmbDiv(re,1);
   int k=1;
   ideal I,delI;
   number g;
   int m=nvars(basering);
//--- prepare the multiplicities of exceptional divisors N and nu
   intvec v=computeV(re,DL);        // nu
   intvec n=computeN(re,DL);        // N
//---------------------------------------------------------------------------
// start computation, first case separately, then loop
//---------------------------------------------------------------------------
   resu[1]=spectral1(1,re,DL,v,n);  // first number, corresponding to
                                    // volume form itself
   if(resu[1]>=bound)
   {
//--- exceeds bound ==> not a spectral number
      resu=delete(resu,1);
      return(resu);
   }
   delI=std(ideal(0));
   while(k)
   {
//--- now run through all monomial x volume form, degree by degree
      j++;
      k=0;
      I=maxideal(j);
      I=reduce(I,delI);
      for(i=1;i<=size(I);i++)
      {
//--- all monomials in degree j
         g=spectral1(I[i],re,DL,v,n);
         if(g<bound)
         {
//--- otherwise g exceeds bound ==> not a spectral number
            k=1;
            l=1;
            while(resu[l]<g)
            {
               l++;
               if(l==size(resu)+1){break;}
            }
            if(l==size(resu)+1){resu[size(resu)+1]=g;}
            if(resu[l]!=g){resu=insert(resu,g,l-1);}
         }
         else
         {
            delI[size(delI)+1]=I[i];
         }
      }
      attrib(delI,"isSB",1);
   }
   return(resu);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=x3+y4+z5;
   list L=resolve(I,"K");
   spectralNeg(L);
   LIB"gmssing.lib";
   ring r=0,(x,y,z),ds;
   poly f=x3+y4+z5;
   spectrum(f);
}

///////////////////////////////////////////////////////////////////////////////
static proc ordE(ideal J,ideal E,ideal W)
"Internal procedure - no help and no example available
"
{
//--- compute multiplicity of E in J -- embedded in W
   int s;
   if(size(J)==0){~;ERROR("ordE: J=0");}
   ideal Estd=std(E+W);
   ideal Epow=1;
   ideal Jquot=1;
   while(size(reduce(Jquot,Estd))!=0)
   {
      s++;
      Epow=Epow*E;
      Jquot=quotient(Epow+W,J);
   }
   return(s-1);
}

///////////////////////////////////////////////////////////////////////////////
proc computeV(list re, list DL)
"USAGE:   computeV(L,DL);
           L = list of rings
          DL = divisor list
ASSUME:   L has structure of output of resolve
          DL has structure of output of prepEmbDiv
RETURN:   intvec,
          i-th entry is multiplicity of i-th divisor in
          pullback of volume form
EXAMPLE:  example computeV;  shows an example
"
{
//--- computes for every divisor E_i its multiplicity + 1 in pi^*(w)
//--- w a non-vanishing 1-form
//--- note: DL is output of prepEmbDiv

//-----------------------------------------------------------------------------
// Initialization
//-----------------------------------------------------------------------------
   def R=basering;
   int i,j,k,n;
   intvec v,w;
   list iden=DL;
   v[size(iden)]=0;

//----------------------------------------------------------------------------
// Run through all exceptional divisors
//----------------------------------------------------------------------------
   for(k=1;k<=size(iden);k++)
   {
      for(i=1;i<=size(iden[k]);i++)
      {
         if(defined(S)){kill S;}
         def S=re[2][iden[k][i][1]];
         setring S;
         if((!v[k])&&(defined(EList)))
         {
            if(defined(II)){kill II;}
//--- we might be embedded in a non-trivial BO[1]
//--- take this into account when forming the jacobi-determinant
            ideal II=jacobDet(BO[5],BO[1]);
            if(size(II)!=0)
            {
                v[k]=ordE(II,EList[iden[k][i][2]],BO[1])+1;
            }
         }
         setring R;
      }
   }
   return(v);
}
example
{"EXAMPLE:"; echo = 2;
  ring R=0,(x,y,z),dp;
  ideal I=(x-y)*(x-z)*(y-z)-z4;
  list re=resolve(I,1);
  list iden=prepEmbDiv(re);
  intvec v=computeV(re, iden);
  v;
}

///////////////////////////////////////////////////////////////////////////////
static proc jacobDet(ideal I, ideal J)
"Internal procedure - no help and no example available
"
{
//--- Returns the Jacobian determinant of the morphism
//--- K[x_1,...,x_m]--->K[y_1,...,y_n]/J defined by x_i ---> I_i.
//--- Let basering=K[y_1,...,y_n], l=n-dim(basering/J),
//--- I=<I_1,...,I_m>, J=<J_1,...,J_r>
//--- For each subset v in {1,...,n} of l elements and
//--- w in {1,...,r} of l elements
//--- let K_v,w be the ideal generated by the n-l-minors of the matrix
//--- (diff(I_i,y_j)+
//---  \sum_k diff(I_i,y_v[k])*diff(J_w[k],y_j))_{j not in v  multiplied with
//---                                 the determinant of (diff(J_w[i],y_v[j]))
//--- the sum of all such ideals K_v,w plus J is returned.

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int n=nvars(basering);
   int i,j,k;
   intvec u,v,w,x;
   matrix MI[ncols(I)][n]=jacob(I);
   matrix N=unitmat(n);
   matrix L;
   ideal K=J;
   if(size(J)==0)
   {
      K=minor(MI,n);
   }

//---------------------------------------------------------------------------
// Do calculation as described above.
// separately for case size(J)=1
//---------------------------------------------------------------------------
   if(size(J)==1)
   {
      matrix MJ[ncols(J)][n]=jacob(J);
      N=concat(N,transpose(MJ));
      v=1..n;
      for(i=1;i<=n;i++)
      {
         L=transpose(permcol(N,i,n+1));
         if(i==1){w=2..n;}
         if(i==n){w=1..n-1;}
         if((i!=1)&&(i!=n)){w=1..i-1,i+1..n;}
         L=submat(L,v,w);
         L=MI*L;
         K=K+minor(L,n-1)*MJ[1,i];
      }
   }
   if(size(J)>1)
   {
      matrix MJ[ncols(J)][n]=jacob(J);
      matrix SMJ;
      N=concat(N,transpose(MJ));
      ideal Jstd=std(J);
      int l=n-dim(Jstd);
      int r=ncols(J);
      list L1=indexSet(n,l);
      list L2=indexSet(r,l);
      for(i=1;i<=size(L1);i++)
      {
         for(j=1;j<=size(L2);j++)
         {
            for(k=1;k<=size(L1[i]);k++)
            {
               if(L1[i][k]){v[size(v)+1]=k;}
            }
            v=v[2..size(v)];
            for(k=1;k<=size(L2[j]);k++)
            {
               if(L2[j][k]){w[size(w)+1]=k;}
            }
            w=w[2..size(w)];
            SMJ=submat(MJ,w,v);
            L=N;
            for(k=1;k<=l;k++)
            {
               L=permcol(L,v[k],n+w[k]);
            }
            u=1..n;
            x=1..n;
            v=sort(v)[1];
            for(k=l;k>=1;k--)
            {
               if(v[k])
               {
                  u=deleteInt(u,v[k],1);
               }
            }
            L=transpose(submat(L,u,x));
            L=MI*L;
            K=K+minor(L,n-l)*det(SMJ);
         }
      }
   }
   return(K);
}

///////////////////////////////////////////////////////////////////////////////
static proc computeH(ideal h,list re,list DL)
"Internal procedure - no help and no example available
"
{
//--- additional procedure to computeV, allows
//--- computation for polynomial x volume form
//--- by computing the contribution of the polynomial h
//--- Note: DL is output of prepEmbDiv

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   def R=basering;
   ideal II=h;
   list iden=DL;
   def T=re[2][1];
   setring T;
   int i,k;
   intvec v;
   v[size(iden)]=0;
   if(deg(II[1])==0){return(v);}
//----------------------------------------------------------------------------
// Run through all exceptional divisors
//----------------------------------------------------------------------------
   for(k=1;k<=size(iden);k++)
   {
      for(i=1;i<=size(iden[k]);i++)
      {
         if(defined(S)){kill S;}
         def S=re[2][iden[k][i][1]];
         setring S;
         if((!v[k])&&(defined(EList)))
         {
            if(defined(JJ)){kill JJ;}
            if(defined(phi)){kill phi;}
            map phi=T,BO[5];
            ideal JJ=phi(II);
            if(size(JJ)!=0)
            {
               v[k]=ordE(JJ,EList[iden[k][i][2]],BO[1]);
            }
         }
         setring R;
      }
   }
   return(v);
}

//////////////////////////////////////////////////////////////////////////////
proc computeN(list re,list DL)
"USAGE:   computeN(L,DL);
          L = list of rings
          DL = divisor list
ASSUME:   L has structure of output of resolve
          DL has structure of output of prepEmbDiv
RETURN:   intvec, i-th entry is multiplicity of i-th divisor
          in total transform under resolution
EXAMPLE:  example computeN;
"
{
//--- computes for every (Q-irred.) divisor E_i its multiplicity in f \circ pi
//--- DL is output of prepEmbDiv

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   def R=basering;
   list iden=DL;
   def T=re[2][1];
   setring T;
   ideal J=BO[2];
   int i,k;
   intvec v;
   v[size(iden)]=0;
//----------------------------------------------------------------------------
// Run through all exceptional divisors
//----------------------------------------------------------------------------
   for(k=1;k<=size(iden);k++)
   {
      for(i=1;i<=size(iden[k]);i++)
      {
         if(defined(S)){kill S;}
         def S=re[2][iden[k][i][1]];
         setring S;
         if((!v[k])&&(defined(EList)))
         {
            if(defined(II)){kill II;}
            if(defined(phi)){kill phi;}
            map phi=T,BO[5];
            ideal II=phi(J);
            if(size(II)!=0)
            {
               v[k]=ordE(II,EList[iden[k][i][2]],BO[1]);
            }
         }
         setring R;
      }
   }
   return(v);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=(x-y)*(x-z)*(y-z)-z4;
   list re=resolve(I,1);
   list iden=prepEmbDiv(re);
   intvec v=computeN(re,iden);
   v;
}
//////////////////////////////////////////////////////////////////////////////
static proc countEijk(list re,list iden,intvec iv,list #)
"Internal procedure - no help and no example available
"
{
//--- count the number of points in the intersection of 3 exceptional
//--- hyperplanes (of dimension 2) - one of them is allowed to be a component
//--- of the strict transform

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,comPa,numPts,localCase;
   intvec ituple,jtuple,ktuple;
   list chList,tmpList;
   def R=basering;
   if(size(#)>0)
   {
      if(string(#[1])=="local")
      {
         localCase=1;
      }
   }
//----------------------------------------------------------------------------
// Find common charts
//----------------------------------------------------------------------------
   for(i=1;i<=size(iden[iv[1]]);i++)
   {
//--- find common charts - only for final charts
      if(defined(S)) {kill S;}
      def S=re[2][iden[iv[1]][i][1]];
      setring S;
      if(!defined(EList))
      {
        i++;
        setring R;
        continue;
      }
      setring R;
      kill ituple,jtuple,ktuple;
      intvec ituple=iden[iv[1]][i];
      intvec jtuple=findInIVList(1,ituple[1],iden[iv[2]]);
      intvec ktuple=findInIVList(1,ituple[1],iden[iv[3]]);
      if((size(jtuple)!=1)&&(size(ktuple)!=1))
      {
//--- chList contains all information about the common charts,
//--- each entry represents a chart and contains three intvecs from iden
//--- one for each E_l
        kill tmpList;
        list tmpList=ituple,jtuple,ktuple;
        chList[size(chList)+1]=tmpList;
        i++;
        if(i<=size(iden[iv[1]]))
        {
           continue;
        }
        else
        {
           break;
        }
      }
   }
   if(size(chList)==0)
   {
//--- no common chart !!!
     return(int(0));
   }
//----------------------------------------------------------------------------
// Count points in common charts
//----------------------------------------------------------------------------
   for(i=1;i<=size(chList);i++)
   {
//--- run through all common charts
      if(defined(S)) { kill S;}
      def S=re[2][chList[i][1][1]];
      setring S;
//--- intersection in this chart
      if(defined(interId)){kill interId;}
      if(localCase==1)
      {
//--- in this case we need to intersect with \pi^-1(0)
         ideal interId=EList[chList[i][1][2]]+EList[chList[i][2][2]]
                                          +EList[chList[i][3][2]]+BO[5];
      }
      else
      {
         ideal interId=EList[chList[i][1][2]]+EList[chList[i][2][2]]
                                          +EList[chList[i][3][2]];
      }
      interId=std(interId);
      if(defined(otherId)) {kill otherId;}
      ideal otherId=1;
      for(j=1;j<i;j++)
      {
//--- run through the previously computed ones
        if(defined(opath)){kill opath;}
        def opath=imap(re[2][chList[j][1][1]],path);
        comPa=1;
        while(opath[1,comPa]==path[1,comPa])
        {
           comPa++;
           if((comPa>ncols(path))||(comPa>ncols(opath))) break;
        }
        comPa=int(leadcoef(path[1,comPa-1]));
        otherId=otherId+interId;
        otherId=intersect(otherId,
                     fetchInTree(re,chList[j][1][1],
                                 comPa,chList[i][1][1],"interId",iden));
      }
      otherId=std(otherId);
//--- do not count each point more than once
      interId=sat(interId,otherId)[1];
      export(interId);
      if(dim(interId)>0)
      {
         ERROR("CountEijk: intersection not zerodimensional");
      }
//--- add the remaining number of points to the total point count numPts
      numPts=numPts+vdim(interId);
   }
   return(numPts);
}
//////////////////////////////////////////////////////////////////////////////
static proc chiEij(list re, list iden, intvec iv)
"Internal procedure - no help and no example available
"
{
//!!! Copy of chiEij_local adjusted for non-local case
//!!! changes must be made in both copies

//--- compute the Euler characteristic of the intersection
//--- curve of two exceptional hypersurfaces (of dimension 2)
//--- one of which is allowed to be a component of the strict transform
//--- using the formula chi(Eij)=2-2g(Eij)

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
  int i,j,k,chi,g;
  intvec ituple,jtuple,inters;
  def R=basering;
//----------------------------------------------------------------------------
// Find a common chart in which they intersect
//----------------------------------------------------------------------------
  for(i=1;i<=size(iden[iv[1]]);i++)
  {
//--- find a common chart in which they intersect: only for final charts
     if(defined(S)) {kill S;}
     def S=re[2][iden[iv[1]][i][1]];
     setring S;
     if(!defined(EList))
     {
       i++;
       setring R;
       continue;
     }
     setring R;
     kill ituple,jtuple;
     intvec ituple=iden[iv[1]][i];
     intvec jtuple=findInIVList(1,ituple[1],iden[iv[2]]);
     if(size(jtuple)==1)
     {
       if(i<size(iden[iv[1]]))
       {
//--- not in this chart
         i++;
         continue;
       }
       else
       {
         if(size(inters)==1)
         {
//--- E_i and E_j do not meet at all
            return("leer");
         }
         else
         {
            return(chi);
         }
       }
     }
//----------------------------------------------------------------------------
// Run through common charts and compute the Euler characteristic of
// each component of Eij.
// As soon as a component has been treated in a chart, it will not be used in
// any subsequent charts.
//----------------------------------------------------------------------------
     if(defined(S)) {kill S;}
     def S=re[2][ituple[1]];
     setring S;
//--- interId: now all components in this chart,
//---          but we want only new components
     if(defined(interId)){kill interId;}
     ideal interId=EList[ituple[2]]+EList[jtuple[2]];
     interId=std(interId);
//--- doneId: already considered components
     if(defined(doneId)){kill doneId;}
     ideal doneId=1;
     for(j=2;j<=size(inters);j++)
     {
//--- fetch the components which have already been dealt with via fetchInTree
        if(defined(opath)) {kill opath;}
        def opath=imap(re[2][inters[j]],path);
        k=1;
        while((k<ncols(opath))&&(k<ncols(path)))
        {
           if(path[1,k+1]!=opath[1,k+1]) break;
           k++;
        }
        if(defined(comPa)) {kill comPa;}
        int comPa=int(leadcoef(path[1,k]));
        if(defined(tempId)){kill tempId;}
        ideal tempId=fetchInTree(re,inters[j],comPa,
                                    iden[iv[1]][i][1],"interId",iden);
        doneId=intersect(doneId,tempId);
        kill tempId;
     }
//--- only consider new components in interId
     interId=sat(interId,doneId)[1];
     if(dim(interId)>1)
     {
       ERROR("genus_Eij: higher dimensional intersection");
     }
     if(dim(interId)>=0)
     {
//--- save the index of the current chart for future use
        export(interId);
        inters[size(inters)+1]=iden[iv[1]][i][1];
     }
     BO[1]=std(BO[1]);
     if(((dim(interId)<=0)&&(dim(BO[1])>2))||
        ((dim(interId)<0)&&(dim(BO[1])==2)))
     {
       if(i<size(iden[iv[1]]))
       {
//--- not in this chart
         setring R;
         i++;
         continue;
       }
       else
       {
         if(size(inters)==1)
         {
//--- E_i and E_j do not meet at all
            return("leer");
         }
         else
         {
            return(chi);
         }
       }
     }
     g=genus(interId);

//--- chi is the Euler characteristic of the (disjoint !!!) union of the
//--- considered components
//--- remark: components are disjoint, because the E_i are normal crossing!!!

     chi=chi+(2-2*g);
  }
  return(chi);
}
//////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////////
static proc chiEij_local(list re, list iden, intvec iv)
"Internal procedure - no help and no example available
"
{
//!!! Copy of chiEij adjusted for local case
//!!! changes must be made in both copies

//--- we have to consider two different cases:
//--- case1: E_i \cap E_j \cap \pi^-1(0) is a curve
//---    compute the Euler characteristic of the intersection
//---    curve of two exceptional hypersurfaces (of dimension 2)
//---    one of which is allowed to be a component of the strict transform
//---    using the formula chi(Eij)=2-2g(Eij)
//--- case2: E_i \cap E_j \cap \pi^-1(0) is a set of points
//---    count the points

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
  int i,j,k,chi,g,points;
  intvec ituple,jtuple,inters;
  def R=basering;
//----------------------------------------------------------------------------
// Find a common chart in which they intersect
//----------------------------------------------------------------------------
  for(i=1;i<=size(iden[iv[1]]);i++)
  {
//--- find a common chart in which they intersect: only for final charts
     if(defined(S)) {kill S;}
     def S=re[2][iden[iv[1]][i][1]];
     setring S;
     if(!defined(EList))
     {
       i++;
       setring R;
       continue;
     }
     setring R;
     kill ituple,jtuple;
     intvec ituple=iden[iv[1]][i];
     intvec jtuple=findInIVList(1,ituple[1],iden[iv[2]]);
     if(size(jtuple)==1)
     {
       if(i<size(iden[iv[1]]))
       {
//--- not in this chart
         i++;
         continue;
       }
       else
       {
         if(size(inters)==1)
         {
//--- E_i and E_j do not meet at all
            return("leer");
         }
         else
         {
            return(chi);
         }
       }
     }
//----------------------------------------------------------------------------
// Run through common charts and compute the Euler characteristic of
// each component of Eij.
// As soon as a component has been treated in a chart, it will not be used in
// any subsequent charts.
//----------------------------------------------------------------------------
     if(defined(S)) {kill S;}
     def S=re[2][ituple[1]];
     setring S;
//--- interId: now all components in this chart,
//---          but we want only new components
     if(defined(interId)){kill interId;}
     ideal interId=EList[ituple[2]]+EList[jtuple[2]]+BO[5];
     interId=std(interId);
//--- doneId: already considered components
     if(defined(doneId)){kill doneId;}
     ideal doneId=1;
     for(j=2;j<=size(inters);j++)
     {
//--- fetch the components which have already been dealt with via fetchInTree
        if(defined(opath)) {kill opath;}
        def opath=imap(re[2][inters[j]],path);
        k=1;
        while((k<ncols(opath))&&(k<ncols(path)))
        {
           if(path[1,k+1]!=opath[1,k+1]) break;
           k++;
        }
        if(defined(comPa)) {kill comPa;}
        int comPa=int(leadcoef(path[1,k]));
        if(defined(tempId)){kill tempId;}
        ideal tempId=fetchInTree(re,inters[j],comPa,
                                    iden[iv[1]][i][1],"interId",iden);
        doneId=intersect(doneId,tempId);
        kill tempId;
     }
//--- only consider new components in interId
     interId=sat(interId,doneId)[1];
     if(dim(interId)>1)
     {
       ERROR("genus_Eij: higher dimensional intersection");
     }
     if(dim(interId)>=0)
     {
//--- save the index of the current chart for future use
        export(interId);
        inters[size(inters)+1]=iden[iv[1]][i][1];
     }
     BO[1]=std(BO[1]);
     if(dim(interId)<0)
     {
       if(i<size(iden[iv[1]]))
       {
//--- not in this chart
         setring R;
         i++;
         continue;
       }
       else
       {
         if(size(inters)==1)
         {
//--- E_i and E_j do not meet at all
            return("leer");
         }
         else
         {
            return(chi);
         }
       }
     }
     if((dim(interId)==0)&&(dim(std(BO[1]))>2))
     {
//--- for sets of points the Euler characteristic is just
//--- the number of points
//--- fat points are impossible, since everything is smooth and n.c.
        chi=chi+vdim(interId);
        points=1;
     }
     else
     {
        if(points==1)
        {
           ERROR("components of intersection do not have same dimension");
        }
        g=genus(interId);
//--- chi is the Euler characteristic of the (disjoint !!!) union of the
//--- considered components
//--- remark: components are disjoint, because the E_i are normal crossing!!!
        chi=chi+(2-2*g);
     }
  }
  return(chi);
}
//////////////////////////////////////////////////////////////////////////////
static proc computeChiE(list re, list iden)
"Internal procedure - no help and no example available
"
{
//--- compute the Euler characteristic of the exceptional hypersurfaces
//--- (of dimension 2), not considering the components of the strict
//--- transform

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,m,thisE,otherE;
   def R=basering;
   intvec nulliv,chi_temp,kvec;
   nulliv[size(iden)]=0;
   list chi_E;
   for(i=1;i<=size(iden);i++)
   {
      chi_E[i]=list();
   }
//---------------------------------------------------------------------------
// Run through the list of charts and compute the Euler characteristic of
// the new exceptional hypersurface and change the values for the old ones
// according to the blow-up which has just been performed
// For initialization reasons, treat the case of the first blow-up separately
//---------------------------------------------------------------------------
   for(i=2;i<=size(re[2]);i++)
   {
//--- run through all charts
      if(defined(S)){kill S;}
      def S=re[2][i];
      setring S;
      m=int(leadcoef(path[1,ncols(path)]));
      if(defined(Spa)){kill Spa;}
      def Spa=re[2][m];
      if(size(BO[4])==1)
      {
//--- just one exceptional divisor
         thisE=1;
         setring Spa;
         if(i==2)
         {
//--- have not set the initial value of chi(E_1) yet
           if(dim(std(cent))==0)
           {
//--- center was point ==> new except. div. is a P^2
              list templist=3*vdim(std(BO[1]+cent)),nulliv;
           }
           else
           {
//--- center was curve ==> new except. div. is curve x P^1
              list templist=4-4*genus(BO[1]+cent),nulliv;
           }
           chi_E[1]=templist;
           kill templist;
         }
         setring S;
         i++;
         if(i<size(re[2]))
         {
            continue;
         }
         else
         {
            break;
         }
      }
      for(j=1;j<=size(iden);j++)
      {
//--- find out which exceptional divisor has just been born
        if(inIVList(intvec(i,size(BO[4])),iden[j]))
        {
//--- found it
           thisE=j;
           break;
        }
      }
//--- now setup new chi and change the previous ones appropriately
      setring Spa;
      if(size(chi_E[thisE])==0)
      {
//--- have not set the initial value of chi(E_thisE) yet
        if(dim(std(cent))==0)
        {
//--- center was point  ==> new except. div. is a P^2
           list templist=3*vdim(std(BO[1]+cent)),nulliv;
        }
        else
        {
//--- center was curve   ==> new except. div. is a C x P^1
           list templist=4-4*genus(BO[1]+cent),nulliv;
        }
        chi_E[thisE]=templist;
        kill templist;
      }
      for(j=1;j<=size(BO[4]);j++)
      {
//--- we are in the parent ring ==> thisE is not yet born
//--- all the other E_i have already been initialized, but the chi
//--- might change with the current blow-up at cent
         if(BO[6][j]==1)
         {
//--- ignore empty sets
            j++;
            if(j<=size(BO[4]))
            {
              continue;
            }
            else
            {
              break;
            }
         }
         for(k=1;k<=size(iden);k++)
         {
//--- find global index of BO[4][j]
            if(inIVList(intvec(m,j),iden[k]))
            {
               otherE=k;
               break;
            }
         }
         if(chi_E[otherE][2][thisE]==1)
         {
//--- already considered this one
            j++;
            if(j<=size(BO[4]))
            {
              continue;
            }
            else
            {
              break;
            }
         }
//---------------------------------------------------------------------------
// update chi according to the formula
// chi(E_k^transf)=chi(E_k) - chi(C \cap E_k) + chi(E_k \cap E_new)
// for convenience of implementation, we first compute
//        chi(E_k) - chi(C \cap E_k)
// and afterwards add the last term chi(E_k \cap E_new)
//---------------------------------------------------------------------------
         ideal CinE=std(cent+BO[4][j]+BO[1]);  // this is C \cap E_k
         if(dim(CinE)==1)
         {
//--- center meets E_k in a curve
            chi_temp[otherE]=chi_E[otherE][1]-(2-2*genus(CinE));
         }
         if(dim(CinE)==0)
         {
//--- center meets E_k in points
            chi_temp[otherE]=chi_E[otherE][1]-vdim(std(CinE));
         }
         kill CinE;
         setring S;
//--- now we are back in the i-th ring in the list
         ideal CinE=std(BO[4][j]+BO[4][size(BO[4])]+BO[1]);
                                              // this is E_k \cap E_new
         if(dim(CinE)==1)
         {
//--- if the two divisors meet, they meet in a curve
            chi_E[otherE][1]=chi_temp[otherE]+(2-2*genus(CinE));
            chi_E[otherE][2][thisE]=1;         // this blow-up of E_k is done
         }
         kill CinE;
         setring Spa;
      }
   }
   setring R;
   return(chi_E);
}
//////////////////////////////////////////////////////////////////////////////
static proc computeChiE_local(list re, list iden)
"Internal procedure - no help and no example available
"
{
//--- compute the Euler characteristic of the intersection of the
//--- exceptional hypersurfaces with \pi^-1(0) which can be of
//--- dimension 1 or 2 - not considering the components of the strict
//--- transform

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,aa,m,n,thisE,otherE;
   def R=basering;
   intvec nulliv,chi_temp,kvec,dimEi,endiv;
   nulliv[size(iden)]=0;
   dimEi[size(iden)]=0;
   endiv[size(re[2])]=0;
   list chi_E;
   for(i=1;i<=size(iden);i++)
   {
      chi_E[i]=list();
   }
//---------------------------------------------------------------------------
// Run through the list of charts and compute the Euler characteristic of
// the new exceptional hypersurface and change the values for the old ones
// according to the blow-up which has just been performed
// For initialization reasons, treat the case of the first blow-up separately
//---------------------------------------------------------------------------
   for(i=2;i<=size(re[2]);i++)
   {
//--- run through all charts
      if(defined(S)){kill S;}
      def S=re[2][i];
      setring S;
      if(defined(EList))
      {
         endiv[i]=1;
      }
      m=int(leadcoef(path[1,ncols(path)]));
      if(defined(Spa)){kill Spa;}
      def Spa=re[2][m];
      if(size(BO[4])==1)
      {
//--- just one exceptional divisor
         thisE=1;
         setring Spa;
         if(i==2)
         {
//--- have not set the initial value of chi(E_1) yet
//--- in the local case, we need to know whether the center contains 0
           if(size(reduce(cent,std(maxideal(1))))!=0)
           {
//--- first center does not meet 0
              list templist=0,nulliv;
              dimEi[1]=-1;
           }
           else
           {
              if(dim(std(cent))==0)
              {
//--- center was point ==> new except. div. is a P^2
                 list templist=3*vdim(std(BO[1]+cent)),nulliv;
                 dimEi[1]=2;
              }
              else
              {
//--- center was curve ==> intersection of new exceptional divisor
//---                      with \pi^-1(0) is a curve, namely P^1
                 setring S;
                 list templist=2,nulliv;
                 dimEi[1]=1;
              }
           }
           chi_E[1]=templist;
           kill templist;
         }
         setring S;
         i++;
         if(i<size(re[2]))
         {
            continue;
         }
         else
         {
            break;
         }
      }
      for(j=1;j<=size(iden);j++)
      {
//--- find out which exceptional divisor has just been born
        if(inIVList(intvec(i,size(BO[4])),iden[j]))
        {
//--- found it
           thisE=j;
           break;
        }
      }
//--- now setup new chi and change the previous ones appropriately
      setring Spa;
      if(size(chi_E[thisE])==0)
      {
//--- have not set the initial value of chi(E_thisE) yet
        if(deg(std(cent+BO[5])[1])==0)
        {
           if(dim(std(cent))==0)
           {
//--- \pi^-1(0) does not meet the Q-point cent
              list templist=0,nulliv;
              dimEi[thisE]=-1;
           }
//--- if cent is a curve, the intersection point might simply be outside
//--- of this chart!!!
        }
        else
        {
           if(dim(std(cent))==0)
           {
//--- center was point  ==> new except. div. is a P^2
              list templist=3*vdim(std(BO[1]+cent)),nulliv;
              dimEi[thisE]=2;
           }
           else
           {
//--- center was curve   ==> new except. div. is a C x P^1
              if(dim(std(cent+BO[5]))==1)
              {
//--- whole curve is in \pi^-1(0)
                 list templist=4-4*genus(BO[1]+cent),nulliv;
                 dimEi[thisE]=2;
              }
              else
              {
//--- curve meets \pi^-1(0) in points
//--- in S, the intersection will be a curve!!!
                 setring S;
                 list templist=2-2*genus(BO[1]+BO[4][size(BO[4])]+BO[5]),nulliv;
                 dimEi[thisE]=1;
                 setring Spa;
              }
           }
        }
        if(defined(templist))
        {
          chi_E[thisE]=templist;
          kill templist;
        }
      }
      for(j=1;j<=size(BO[4]);j++)
      {
//--- we are in the parent ring ==> thisE is not yet born
//--- all the other E_i have already been initialized, but the chi
//--- might change with the current blow-up at cent
         if(BO[6][j]==1)
         {
//--- ignore empty sets
            j++;
            if(j<=size(BO[4]))
            {
              continue;
            }
            else
            {
              break;
            }
         }
         for(k=1;k<=size(iden);k++)
         {
//--- find global index of BO[4][j]
            if(inIVList(intvec(m,j),iden[k]))
            {
               otherE=k;
               break;
            }
         }
         if(dimEi[otherE]<=1)
         {
//--- dimEi[otherE]==-1: center leading to this E does not meet \pi^-1(0)
//--- dimEi[otherE]== 0: center leading to this E does not meet \pi^-1(0)
//---                   in any previously visited charts
//---                   maybe in some other branch later, but has nothing
//---                   to do with this center
//--- dimEi[otherE]== 1: E \cap \pi^-1(0) is curve
//---                   ==> chi is birational invariant
            j++;
            if(j<=size(BO[4]))
            {
              continue;
            }
            break;
         }
         if(chi_E[otherE][2][thisE]==1)
         {
//--- already considered this one
            j++;
            if(j<=size(BO[4]))
            {
              continue;
            }
            else
            {
              break;
            }
         }
//---------------------------------------------------------------------------
// update chi according to the formula
// chi(E_k^transf)=chi(E_k) - chi(C \cap E_k) + chi(E_k \cap E_new)
// for convenience of implementation, we first compute
//        chi(E_k) - chi(C \cap E_k)
// and afterwards add the last term chi(E_k \cap E_new)
//---------------------------------------------------------------------------
         ideal CinE=std(cent+BO[4][j]+BO[1]);  // this is C \cap E_k
         if(dim(CinE)==1)
         {
//--- center meets E_k in a curve
            chi_temp[otherE]=chi_E[otherE][1]-(2-2*genus(CinE));
         }
         if(dim(CinE)==0)
         {
//--- center meets E_k in points
            chi_temp[otherE]=chi_E[otherE][1]-vdim(std(CinE));
         }
         kill CinE;
         setring S;
//--- now we are back in the i-th ring in the list
         ideal CinE=std(BO[4][j]+BO[4][size(BO[4])]+BO[1]);
                                              // this is E_k \cap E_new
         if(dim(CinE)==1)
         {
//--- if the two divisors meet, they meet in a curve
            chi_E[otherE][1]=chi_temp[otherE][1]+(2-2*genus(CinE));
            chi_E[otherE][2][thisE]=1;         // this blow-up of E_k is done
         }
         kill CinE;
         setring Spa;
      }
   }
//--- we still need to clean-up the 1-dimensional E_i \cap \pi^-1(0)
   for(i=1;i<=size(dimEi);i++)
   {
      if(dimEi[i]!=1)
      {
//--- not 1-dimensional ==> skip
         i++;
         if(i>size(dimEi)) break;
         continue;
      }
      if(defined(myCharts)) {kill myCharts;}
      intvec myCharts;
      chi_E[i]=0;
      for(j=1;j<=size(re[2]);j++)
      {
         if(endiv[j]==0)
         {
//--- not an endChart ==> skip
            j++;
            if(j>size(re[2])) break;
            continue;
         }
         if(defined(mtuple)) {kill mtuple;}
         intvec mtuple=findInIVList(1,j,iden[i]);
         if(size(mtuple)==1)
         {
//-- nothing to do with this Ei ==> skip
            j++;
            if(j>size(re[2])) break;
            continue;
         }
         myCharts[size(myCharts)+1]=j;
         if(defined(S)){kill S;}
         def S=re[2][j];
         setring S;
         if(defined(interId)){kill interId;}
//--- all components
         ideal interId=std(BO[4][mtuple[2]]+BO[5]);
         if(defined(myPts)){kill myPts;}
         ideal myPts=1;
         export(myPts);
         export(interId);
         if(defined(doneId)){kill doneId;}
         if(defined(donePts)){kill donePts;}
         ideal donePts=1;
         ideal doneId=1;
         for(k=2;k<size(myCharts);k++)
         {
//--- fetch the components which have already been dealt with via fetchInTree
            if(defined(opath)) {kill opath;}
            def opath=imap(re[2][myCharts[k][1]],path);
            aa=1;
            while((aa<ncols(opath))&&(aa<ncols(path)))
            {
               if(path[1,aa+1]!=opath[1,aa+1]) break;
               aa++;
            }
            if(defined(comPa)) {kill comPa;}
            int comPa=int(leadcoef(path[1,aa]));
            if(defined(tempId)){kill tempId;}
            ideal tempId=fetchInTree(re,myCharts[k][1],comPa,j,"interId",iden);
            doneId=intersect(doneId,tempId);
            kill tempId;
            ideal tempId=fetchInTree(re,myCharts[k][1],comPa,j,"myPts",iden);
            donePts=intersect(donePts,tempId);
            kill tempId;
         }
//--- drop components which have already been dealt with
         interId=sat(interId,doneId)[1];
         list pr=minAssGTZ(interId);
         myPts=std(interId+doneId);
         for(k=1;k<=size(pr);k++)
         {
           for(n=k+1;n<=size(pr);n++)
           {
              myPts=intersect(myPts,std(pr[k]+pr[n]));
           }
           if(deg(std(pr[k])[1])>0)
           {
              chi_E[i][1]=chi_E[i][1]+(2-2*genus(pr[k]));
           }
         }
         myPts=sat(myPts,donePts)[1];
         chi_E[i][1]=chi_E[i][1]-vdim(myPts);
      }
   }
   setring R;
   return(chi_E);
}
//////////////////////////////////////////////////////////////////////////////
static proc chi_ast(list re,list iden,list #)
"Internal procedure - no help and no example available
"
{
//--- compute the Euler characteristic of the Ei,Eij,Eijk and the
//--- corresponding Ei^*,Eij^*,Eijk^* by preparing the input to the
//--- specialized auxilliary procedures and then recombining the results

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,g;
   intvec tiv;
   list chi_ijk,chi_ij,chi_i,ast_ijk,ast_ij,ast_i,tmplist,g_ij,emptylist;
   list leererSchnitt;
   def R=basering;
   ring Rhelp=0,@t,dp;
   setring R;
//----------------------------------------------------------------------------
// first compute the chi(Eij) and  at the same time
// check whether E_i \cap E_j is empty
// the formula is
//              chi_ij=2-2*genus(E_i \cap E_j)
//----------------------------------------------------------------------------
   if(size(#)>0)
   {
      "Entering chi_ast";
   }
   for(i=1;i<=size(iden)-1;i++)
   {
      for(j=i+1;j<=size(iden);j++)
      {
         if(defined(blub)){kill blub;}
         def blub=chiEij(re,iden,intvec(i,j));
         if(typeof(blub)=="int")
         {
            tmplist=intvec(i,j),blub;
         }
         else
         {
            leererSchnitt[size(leererSchnitt)+1]=intvec(i,j);
            tmplist=intvec(i,j),0;
         }
         chi_ij[size(chi_ij)+1]=tmplist;
      }
   }
   if(size(#)>0)
   {
      "chi_ij computed";
   }
//-----------------------------------------------------------------------------
// compute chi(Eijk)=chi^*(Eijk) by counting the points in the intersection
//      chi_ijk=#(E_i \cap E_j \cap E_k)
//      ast_ijk=chi_ijk
//-----------------------------------------------------------------------------
   for(i=1;i<=size(iden)-2;i++)
   {
      for(j=i+1;j<=size(iden)-1;j++)
      {
         for(k=j+1;k<=size(iden);k++)
         {
            if(inIVList(intvec(i,j),leererSchnitt))
            {
               tmplist=intvec(i,j,k),0;
            }
            else
            {
               tmplist=intvec(i,j,k),countEijk(re,iden,intvec(i,j,k));
            }
            chi_ijk[size(chi_ijk)+1]=tmplist;
         }
      }
   }
   ast_ijk=chi_ijk;
   if(size(#)>0)
   {
      "chi_ijk computed";
   }
//----------------------------------------------------------------------------
// construct chi(Eij^*) by the formula
//      ast_ij=chi_ij - sum_ijk chi_ijk,
// where k runs over all indices != i,j
//----------------------------------------------------------------------------
   for(i=1;i<=size(chi_ij);i++)
   {
      ast_ij[i]=chi_ij[i];
      for(k=1;k<=size(chi_ijk);k++)
      {
         if(((chi_ijk[k][1][1]==chi_ij[i][1][1])||
                 (chi_ijk[k][1][2]==chi_ij[i][1][1]))&&
            ((chi_ijk[k][1][2]==chi_ij[i][1][2])||
                 (chi_ijk[k][1][3]==chi_ij[i][1][2])))
         {
            ast_ij[i][2]=ast_ij[i][2]-chi_ijk[k][2];
         }
      }
   }
   if(size(#)>0)
   {
      "ast_ij computed";
   }
//----------------------------------------------------------------------------
// construct ast_i according to the following formulae
//      ast_i=0 if E_i is (Q- resp. C-)component of the strict transform
//      chi_i=3*n if E_i originates from blowing up a Q-point,
//                which consists of n (different) C-points
//      chi_i=2-2g(C) if E_i originates from blowing up a (Q-)curve C
//           (chi_i=n*(2-2g(C_i))=2-2g(C),
//                 where C=\cup C_i, C_i \cap C_j = \emptyset)
// if E_i is not a component of the strict transform, then
//       ast_i=chi_i - sum_{j!=i} ast_ij
//----------------------------------------------------------------------------
   for(i=1;i<=size(iden);i++)
   {
      if(defined(S)) {kill S;}
      def S=re[2][iden[i][1][1]];
      setring S;
      if(iden[i][1][2]>size(BO[4]))
      {
         i--;
         break;
      }
   }
   list idenE=iden;
   while(size(idenE)>i)
   {
      idenE=delete(idenE,size(idenE));
   }
   list cl=computeChiE(re,idenE);
   for(i=1;i<=size(idenE);i++)
   {
      chi_i[i]=list(intvec(i),cl[i][1]);
   }
   if(size(#)>0)
   {
      "chi_i computed";
   }
   for(i=1;i<=size(idenE);i++)
   {
      ast_i[i]=chi_i[i];
      for(j=1;j<=size(ast_ij);j++)
      {
         if((ast_ij[j][1][1]==i)||(ast_ij[j][1][2]==i))
         {
            ast_i[i][2]=ast_i[i][2]-chi_ij[j][2];
         }
      }
      for(j=1;j<=size(ast_ijk);j++)
      {
         if((ast_ijk[j][1][1]==i)||(ast_ijk[j][1][2]==i)
                                 ||(ast_ijk[j][1][3]==i))
         {
            ast_i[i][2]=ast_i[i][2]+chi_ijk[j][2];
         }
      }
   }
   for(i=size(idenE)+1;i<=size(iden);i++)
   {
      ast_i[i]=list(intvec(i),0);
   }
//--- results are in ast_i, ast_ij and ast_ijk
//--- all are of the form intvec(indices),int(value)
   list result=ast_i,ast_ij,ast_ijk;
   return(result);
}
//////////////////////////////////////////////////////////////////////////////
static proc chi_ast_local(list re,list iden,list #)
"Internal procedure - no help and no example available
"
{
//--- compute the Euler characteristic of the Ei,Eij,Eijk and the
//--- corresponding Ei^*,Eij^*,Eijk^* by preparing the input to the
//--- specialized auxilliary procedures and then recombining the results

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,g;
   intvec tiv;
   list chi_ijk,chi_ij,chi_i,ast_ijk,ast_ij,ast_i,tmplist,g_ij,emptylist;
   list leererSchnitt;
   def R=basering;
   ring Rhelp=0,@t,dp;
   setring R;
//----------------------------------------------------------------------------
// first compute
// if E_i \cap E_j \cap \pi^-1(0) is a curve:
//    chi(Eij) and  at the same time
//    check whether E_i \cap E_j is empty
//    the formula is
//              chi_ij=2-2*genus(E_i \cap E_j)
// otherwise (points):
//    chi(E_ij) by counting the points
//----------------------------------------------------------------------------
   if(size(#)>0)
   {
      "Entering chi_ast_local";
   }
   for(i=1;i<=size(iden)-1;i++)
   {
      for(j=i+1;j<=size(iden);j++)
      {
         if(defined(blub)){kill blub;}
         def blub=chiEij_local(re,iden,intvec(i,j));
         if(typeof(blub)=="int")
         {
            tmplist=intvec(i,j),blub;
         }
         else
         {
            leererSchnitt[size(leererSchnitt)+1]=intvec(i,j);
            tmplist=intvec(i,j),0;
         }
         chi_ij[size(chi_ij)+1]=tmplist;
      }
   }
   if(size(#)>0)
   {
      "chi_ij computed";
   }
//-----------------------------------------------------------------------------
// compute chi(Eijk)=chi^*(Eijk) by counting the points in the intersection
//      chi_ijk=#(E_i \cap E_j \cap E_k \cap \pi^-1(0))
//      ast_ijk=chi_ijk
//-----------------------------------------------------------------------------
   for(i=1;i<=size(iden)-2;i++)
   {
      for(j=i+1;j<=size(iden)-1;j++)
      {
         for(k=j+1;k<=size(iden);k++)
         {
            if(inIVList(intvec(i,j),leererSchnitt))
            {
               tmplist=intvec(i,j,k),0;
            }
            else
            {
               tmplist=intvec(i,j,k),countEijk(re,iden,intvec(i,j,k),"local");
            }
            chi_ijk[size(chi_ijk)+1]=tmplist;
         }
      }
   }
   ast_ijk=chi_ijk;
   if(size(#)>0)
   {
      "chi_ijk computed";
   }
//----------------------------------------------------------------------------
// construct chi(Eij^*) by the formula
//      ast_ij=chi_ij - sum_ijk chi_ijk,
// where k runs over all indices != i,j
//----------------------------------------------------------------------------
   for(i=1;i<=size(chi_ij);i++)
   {
      ast_ij[i]=chi_ij[i];
      for(k=1;k<=size(chi_ijk);k++)
      {
         if(((chi_ijk[k][1][1]==chi_ij[i][1][1])||
                 (chi_ijk[k][1][2]==chi_ij[i][1][1]))&&
            ((chi_ijk[k][1][2]==chi_ij[i][1][2])||
                 (chi_ijk[k][1][3]==chi_ij[i][1][2])))
         {
            ast_ij[i][2]=ast_ij[i][2]-chi_ijk[k][2];
         }
      }
   }
   if(size(#)>0)
   {
      "ast_ij computed";
   }
//----------------------------------------------------------------------------
// construct ast_i according to the following formulae
//      ast_i=0 if E_i is (Q- resp. C-)component of the strict transform
// if E_i \cap \pi^-1(0) is of dimension 2:
//      chi_i=3*n if E_i originates from blowing up a Q-point,
//                which consists of n (different) C-points
//      chi_i=2-2g(C) if E_i originates from blowing up a (Q-)curve C
//           (chi_i=n*(2-2g(C_i))=2-2g(C),
//                 where C=\cup C_i, C_i \cap C_j = \emptyset)
// if E_i \cap \pi^-1(0) is a curve:
//      use the formula chi_i=2-2*genus(E_i \cap \pi^-1(0))
//
// for E_i not a component of the strict transform we have
//       ast_i=chi_i - sum_{j!=i} ast_ij
//----------------------------------------------------------------------------
   for(i=1;i<=size(iden);i++)
   {
      if(defined(S)) {kill S;}
      def S=re[2][iden[i][1][1]];
      setring S;
      if(iden[i][1][2]>size(BO[4]))
      {
         i--;
         break;
      }
   }
   list idenE=iden;
   while(size(idenE)>i)
   {
      idenE=delete(idenE,size(idenE));
   }
   list cl=computeChiE_local(re,idenE);
   for(i=1;i<=size(cl);i++)
   {
      if(size(cl[i])==0)
      {
         cl[i][1]=0;
      }
   }
   for(i=1;i<=size(idenE);i++)
   {
      chi_i[i]=list(intvec(i),cl[i][1]);
   }
   if(size(#)>0)
   {
      "chi_i computed";
   }
   for(i=1;i<=size(idenE);i++)
   {
      ast_i[i]=chi_i[i];
      for(j=1;j<=size(ast_ij);j++)
      {
         if((ast_ij[j][1][1]==i)||(ast_ij[j][1][2]==i))
         {
            ast_i[i][2]=ast_i[i][2]-chi_ij[j][2];
         }
      }
      for(j=1;j<=size(ast_ijk);j++)
      {
         if((ast_ijk[j][1][1]==i)||(ast_ijk[j][1][2]==i)
                                 ||(ast_ijk[j][1][3]==i))
         {
            ast_i[i][2]=ast_i[i][2]+chi_ijk[j][2];
         }
      }
   }
   for(i=size(idenE)+1;i<=size(iden);i++)
   {
      ast_i[i]=list(intvec(i),0);
   }
//--- results are in ast_i, ast_ij and ast_ijk
//--- all are of the form intvec(indices),int(value)
//"End of chi_ast_local";
//~;
   list result=ast_i,ast_ij,ast_ijk;
   return(result);
}
//////////////////////////////////////////////////////////////////////////////

proc discrepancy(list re)
"USAGE:   discrepancy(L);
@*        L    = list of rings
ASSUME:   L is the output of resolution of singularities
RETRUN:   discrepancies of the given resolution"
{
//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   def R=basering;
   int i,j;
   list iden=prepEmbDiv(re);       //--- identify the E_i
   intvec Vvec=computeV(re,iden);  //--- nu
   intvec Nvec=computeN(re,iden);  //--- N
   intvec Avec;
//--- only look at exceptional divisors, not at strict transform
   for(i=1;i<=size(iden);i++)
   {
      if(defined(S)) {kill S;}
      def S=re[2][iden[i][1][1]];
      setring S;
      if(iden[i][1][2]>size(BO[4]))
      {
         i--;
         break;
      }
   }
   j=i;
//--- discrepancies are a_i=nu_i-N_i
   for(i=1;i<=j;i++)
   {
     Avec[i]=Vvec[i]-Nvec[i]-1;
   }
   return(Avec);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=x2+y2+z3;
   list re=resolve(I);
   discrepancy(re);
}
//////////////////////////////////////////////////////////////////////////////

proc zetaDL(list re,int d,list #)
"USAGE:   zetaDL(L,d[,s1][,s2][,a]);
          L     = list of rings;
          d     = integer;
          s1,s2 = string;
          a     = integer
ASSUME:   L is the output of resolution of singularities
COMPUTE:  local Denef-Loeser zeta function, if string s1 is present and
             has the value 'local'; global Denef-Loeser zeta function
             otherwise
          if string s1 or s2 has the value "A", additionally the
             characteristic polynomial of the monodromy is computed
RETURN:   list l
          if a is not present:
          l[1]: string specifying the top. zeta function
          l[2]: string specifying characteristic polynomial of monodromy,
                if "A" was specified
          if a is present:
          l[1]: string specifying the top. zeta function
          l[2]: list ast,
                     ast[1]=chi(Ei^*)
                     ast[2]=chi(Eij^*)
                     ast[3]=chi(Eijk^*)
          l[3]: intvec nu of multiplicites as needed in computation of zeta
                function
          l[4]: intvec N of multiplicities as needed in compuation of zeta
                function
          l[5]: string specifying characteristic polynomial of monodromy,
                if "A" was specified
EXAMPLE:  example zetaDL;   shows an example
"
{
//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   def R=basering;
   int show_all,i;
   if(size(#)>0)
   {
     if((typeof(#[1])=="int")||(size(#)>2))
     {
       show_all=1;
     }
     if(typeof(#[1])=="string")
     {
       if((#[1]=="local")||(#[1]=="lokal"))
       {
//          ERROR("Local case not implemented yet");
          "Local Case: Assuming that no (!) charts were dropped";
          "during calculation of the resolution (option \"A\")";
          int localComp=1;
          if(size(#)>1)
          {
             if(#[2]=="A")
             {
                int aCampoFormula=1;
             }
          }
       }
       else
       {
          if(#[1]=="A")
          {
             int aCampoFormula=1;
          }
          "Computing global zeta function";
       }
     }
   }
//----------------------------------------------------------------------------
// Identify the embedded divisors and chi(Ei^*), chi(Eij^*) and chi(Eijk^*)
// as well as the integer vector V(=nu) and N
//----------------------------------------------------------------------------
   list iden=prepEmbDiv(re);       //--- identify the E_i
//!!! TIMING:    E8 takes 520 sec ==> needs speed up
   if(!defined(localComp))
   {
      list ast_list=chi_ast(re,iden); //--- compute chi(E^*)
   }
   else
   {
      list ast_list=chi_ast_local(re,iden);
   }
   intvec Vvec=computeV(re,iden);  //--- nu
   intvec Nvec=computeN(re,iden);  //--- N
//----------------------------------------------------------------------------
// Build a new ring with one parameter s
// and compute Zeta_top^(d) in its ground field
//----------------------------------------------------------------------------
   ring Qs=(0,s),x,dp;
   number zetaTop=0;
   number enum,denom;
   denom=1;
   for(i=1;i<=size(Nvec);i++)
   {
      denom=denom*(Vvec[i]+s*Nvec[i]);
   }
//--- factors for which index set J consists of one element
//--- (do something only if d divides N_j)
   for(i=1;i<=size(ast_list[1]);i++)
   {
      if((((Nvec[ast_list[1][i][1][1]] div d)*d)-Nvec[ast_list[1][i][1][1]]==0)&&
         (ast_list[1][i][2]!=0))
      {
          enum=enum+ast_list[1][i][2]*(denom/(Vvec[ast_list[1][i][1][1]]+s*Nvec[ast_list[1][i][1][1]]));
      }
   }
//--- factors for which index set J consists of two elements
//--- (do something only if d divides both N_i and N_j)
//!!! TIMING: E8 takes 690 sec and has 703 elements
//!!!         ==> need to implement a smarter way to do this
//!!! e.g. build up enumerator and denominator separately, thus not
//!!!      searching for common factors in each step
   for(i=1;i<=size(ast_list[2]);i++)
   {
      if((((Nvec[ast_list[2][i][1][1]] div d)*d)-Nvec[ast_list[2][i][1][1]]==0)&&
         (((Nvec[ast_list[2][i][1][2]] div d)*d)-Nvec[ast_list[2][i][1][2]]==0)&&
          (ast_list[2][i][2]!=0))
      {
         enum=enum+ast_list[2][i][2]*(denom/((Vvec[ast_list[2][i][1][1]]+s*Nvec[ast_list[2][i][1][1]])*(Vvec[ast_list[2][i][1][2]]+s*Nvec[ast_list[2][i][1][2]])));
      }
   }
//--- factors for which index set J consists of three elements
//--- (do something only if d divides N_i, N_j and N_k)
//!!! TIMING: E8 takes 490 sec and has 8436 elements
//!!!         ==> same kind of improvements as in the previous case needed
   for(i=1;i<=size(ast_list[3]);i++)
   {
      if((((Nvec[ast_list[3][i][1][1]] div d)*d)-Nvec[ast_list[3][i][1][1]]==0)&&
         (((Nvec[ast_list[3][i][1][2]] div d)*d)-Nvec[ast_list[3][i][1][2]]==0)&&
         (((Nvec[ast_list[3][i][1][3]] div d)*d)-Nvec[ast_list[3][i][1][3]]==0)&&
         (ast_list[3][i][2]!=0))
      {
         enum=enum+ast_list[3][i][2]*(denom/((Vvec[ast_list[3][i][1][1]]+s*Nvec[ast_list[3][i][1][1]])*(Vvec[ast_list[3][i][1][2]]+s*Nvec[ast_list[3][i][1][2]])*(Vvec[ast_list[3][i][1][3]]+s*Nvec[ast_list[3][i][1][3]])));
      }
   }
   zetaTop=enum/denom;
   zetaTop=numerator(zetaTop)/denominator(zetaTop);
   string zetaStr=string(zetaTop);

   if(show_all)
   {
      list result=zetaStr,ast_list[1],ast_list[2],ast_list[3],Vvec,Nvec;
   }
   else
   {
      list result=zetaStr;
   }
//--- compute characteristic polynomial of the monodromy
//--- by the A'Campo formula
   if(defined(aCampoFormula))
   {
      poly charP=1;
      for(i=1;i<=size(ast_list[1]);i++)
      {
         charP=charP*((s^Nvec[i]-1)^ast_list[1][i][2]);
      }
      string charPStr=string(charP/(s-1));
      result[size(result)+1]=charPStr;
   }
   setring R;
   return(result);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=x2+y2+z3;
   list re=resolve(I,"K");
   zetaDL(re,1);
   I=(xz+y2)*(xz+y2+x2)+z5;
   list L=resolve(I,"K");
   zetaDL(L,1);

//===== expected zeta function =========
// (20s^2+130s+87)/((1+s)*(3+4s)*(29+40s))
//======================================
}
//////////////////////////////////////////////////////////////////////////////

proc abstractR(list re)
"USAGE:   abstractR(L);
@*        L = list of rings
ASSUME:   L is output of resolution of singularities
NOTE:     currently only implemented for isolated surface singularities
RETURN:   list l
          l[1]: intvec, where
                   l[1][i]=1 if the corresponding ring is a final chart
                             of non-embedded resolution
                   l[1][i]=0 otherwise
          l[2]: intvec, where
                   l[2][i]=1 if the corresponding ring does not occur
                             in the non-embedded resolution
                   l[2][i]=0 otherwise
          l[3]: list L
EXAMPLE:  example abstractR;   shows an example
"
{
//---------------------------------------------------------------------------
// Initialization and sanity checks
//---------------------------------------------------------------------------
   def R=basering;
//---Test whether we are in the irreducible surface case
   def S=re[2][1];
   setring S;
   BO[2]=BO[2]+BO[1];
   if(dim(std(BO[2]))!=2)
   {
       ERROR("NOT A SURFACE");
   }
   if(dim(std(slocus(BO[2])))>0)
   {
       ERROR("NOT AN ISOLATED SINGULARITY");
   }
   setring R;
   int i,j,k,l,i0;
   intvec deleted;
   intvec endiv;
   endiv[size(re[2])]=0;
   deleted[size(re[2])]=0;
//-----------------------------------------------------------------------------
// run through all rings, only consider final charts
// for each final chart follow the list of charts leading up to it until
//     we encounter a chart which is not finished in the non-embedded case
//-----------------------------------------------------------------------------
   for(i=1;i<=size(re[2]);i++)
   {
      if(defined(S)){kill S;}
      def S=re[2][i];
      setring S;
      if(size(reduce(cent,std(BO[2]+BO[1])))!=0)
      {
//--- only consider endrings
         i++;
         continue;
      }
      i0=i;
      for(j=ncols(path);j>=2;j--)
      {
//--- walk backwards through history
        if(j==2)
        {
          endiv[i0]=1;
          break;
        }
        k=int(leadcoef(path[1,j]));
        if((deleted[k]==1)||(endiv[k]==1))
        {
          deleted[i0]=1;
          break;
        }
        if(defined(SPa)){kill SPa;}
        def SPa=re[2][k];
        setring SPa;
        l=int(leadcoef(path[1,ncols(path)]));
        if(defined(SPa2)){kill SPa2;}
        def SPa2=re[2][l];
        setring SPa2;
        if((deleted[l]==1)||(endiv[l]==1))
        {
//--- parent was already treated via different final chart
//--- we may safely inherit the data
          deleted[i0]=1;
          setring S;
          i0=k;
          j--;
          continue;
        }
        setring SPa;
//!!! Idea of Improvement:
//!!! BESSER: rueckwaerts gehend nur testen ob glatt
//!!!         danach vorwaerts bis zum ersten Mal abstractNC
//!!! ACHTUNG: rueckweg unterwegs notieren - wir haben nur vergangenheit!
        if((deg(std(slocus(BO[2]))[1])!=0)||(!abstractNC(BO)))
        {
//--- not finished in the non-embedded case
           endiv[i0]=1;
           break;
        }
//--- unnecessary chart in non-embedded case
        setring S;
        deleted[i0]=1;
        i0=k;
      }
   }
//-----------------------------------------------------------------------------
// Clean up the intvec deleted and return the result
//-----------------------------------------------------------------------------
   setring R;
   for(i=1;i<=size(endiv);i++)
   {
     if(endiv[i]==1)
     {
        if(defined(S)) {kill S;}
        def S=re[2][i];
        setring S;
        for(j=3;j<ncols(path);j++)
        {
           if((endiv[int(leadcoef(path[1,j]))]==1)||
              (deleted[int(leadcoef(path[1,j]))]==1))
           {
              deleted[int(leadcoef(path[1,j+1]))]=1;
              endiv[int(leadcoef(path[1,j+1]))]=0;
           }
        }
        if((endiv[int(leadcoef(path[1,ncols(path)]))]==1)||
           (deleted[int(leadcoef(path[1,ncols(path)]))]==1))
        {
           deleted[i]=1;
           endiv[i]=0;
        }
     }
   }
   list resu=endiv,deleted,re;
   return(resu);
}
example
{"EXAMPLE:";
   echo = 2;
   ring r = 0,(x,y,z),dp;
   ideal I=x2+y2+z11;
   list L=resolve(I);
   list absR=abstractR(L);
   absR[1];
   absR[2];
}
//////////////////////////////////////////////////////////////////////////////
static proc decompE(list BO)
"Internal procedure - no help and no example available
"
{
//--- compute the list of exceptional divisors, including the components
//--- of the strict transform in the non-embedded case
//--- (computation over Q !!!)
   def R=basering;
   list Elist,prList;
   int i;
   for(i=1;i<=size(BO[4]);i++)
   {
      Elist[i]=BO[4][i];
   }
/* practical speed up (part 1 of 3) -- no theoretical relevance
   ideal M=maxideal(1);
   M[1]=var(nvars(basering));
   M[nvars(basering)]=var(1);
   map phi=R,M;
*/
   ideal KK=BO[2];

/* practical speed up (part 2 of 3)
   KK=phi(KK);
*/
   prList=minAssGTZ(KK);

/* practical speed up (part 3 of 3)
   prList=phi(prList);
*/

   for(i=1;i<=size(prList);i++)
   {
      Elist[size(BO[4])+i]=prList[i];
   }
   return(Elist);
}
//////////////////////////////////////////////////////////////////////////////

proc prepEmbDiv(list re, list #)
"USAGE:   prepEmbDiv(L[,a]);
@*        L = list of rings
@*        a = integer
ASSUME:   L is output of resolution of singularities
COMPUTE:  if a is not present: exceptional divisors including components
          of the strict transform
          otherwise: only exceptional divisors
RETURN:   list of Q-irreducible exceptional divisors (embedded case)
EXAMPLE:  example prepEmbDiv;   shows an example
"
{
//--- 1) in each final chart, a list of (decomposed) exceptional divisors
//---    is created (and exported)
//--- 2) the strict transform is decomposed
//--- 3) the exceptional divisors (including the strict transform)
//---    in the different charts are compared, identified and this
//---    information collected into a list which is then returned
//---------------------------------------------------------------------------
// Initialization
//---------------------------------------------------------------------------
   int i,j,k,ncomps,offset,found,a,b,c,d;
   list tmpList;
   def R=basering;
//--- identify identical exceptional divisors
//--- (note: we are in the embedded case)
   list iden=collectDiv(re)[2];
//---------------------------------------------------------------------------
// Go to each final chart and create the EList
//---------------------------------------------------------------------------
   for(i=1;i<=size(iden[size(iden)]);i++)
   {
      if(defined(S)){kill S;}
      def S=re[2][iden[size(iden)][i][1]];
      setring S;
      if(defined(EList)){kill EList;}
      list EList=decompE(BO);
      export(EList);
      setring R;
      kill S;
   }
//--- save original iden for further use and then drop
//--- strict transform from it
   list iden0=iden;
   iden=delete(iden,size(iden));
   if(size(#)>0)
   {
//--- we are not interested in the strict transform of X
      return(iden);
   }
//----------------------------------------------------------------------------
// Run through all final charts and collect and identify all components of
// the strict transform
//----------------------------------------------------------------------------
//--- first final chart - to be used for initialization
   def S=re[2][iden0[size(iden0)][1][1]];
   setring S;
   ncomps=size(EList)-size(BO[4]);
   if((ncomps==1)&&(deg(std(EList[size(EList)])[1])==0))
   {
      ncomps=0;
   }
   offset=size(BO[4]);
   for(i=1;i<=ncomps;i++)
   {
//--- add components of strict transform
      tmpList[1]=intvec(iden0[size(iden0)][1][1],size(BO[4])+i);
      iden[size(iden)+1]=tmpList;
   }
//--- now run through the other final charts
   for(i=2;i<=size(iden0[size(iden0)]);i++)
   {
      if(defined(S2)){kill S2;}
      def S2=re[2][iden0[size(iden0)][i][1]];
      setring S2;
//--- determine common parent of this ring and re[2][iden0[size(iden0)][1][1]]
      if(defined(opath)){kill opath;}
      def opath=imap(S,path);
      j=1;
      while(opath[1,j]==path[1,j])
      {
         j++;
         if((j>ncols(path))||(j>ncols(opath))) break;
      }
      if(defined(li1)){kill li1;}
      list li1;
//--- fetch the components we have considered in
//--- re[2][iden0[size(iden0)][1][1]]
//--- via the resolution tree
      for(k=1;k<=ncomps;k++)
      {
         if(defined(id1)){kill id1;}
         string tempstr="EList["+string(eval(k+offset))+"]";
         ideal id1=fetchInTree(re,iden0[size(iden0)][1][1],
                               int(leadcoef(path[1,j-1])),
                               iden0[size(iden0)][i][1],tempstr,iden0,1);
         kill tempstr;
         li1[k]=id1;
         kill id1;
      }
//--- do the comparison
      for(k=size(BO[4])+1;k<=size(EList);k++)
      {
//--- only components of the strict transform are interesting
         if((size(BO[4])+1==size(EList))&&(deg(std(EList[size(EList)])[1])==0))
         {
            break;
         }
         found=0;
         for(j=1;j<=size(li1);j++)
         {
            if((size(reduce(li1[j],std(EList[k])))==0)&&
               (size(reduce(EList[k],std(li1[j])))==0))
            {
//--- found a match
               li1[j]=ideal(1);
               iden[size(iden0)-1+j][size(iden[size(iden0)-1+j])+1]=
                      intvec(iden0[size(iden0)][i][1],k);
               found=1;
               break;
            }
         }
         if(!found)
         {
//--- no match yet, maybe there are entries not corresponding to the
//--- initialization of the list -- collected in list repair
            if(!defined(repair))
            {
//--- no entries in repair, we add the very first one
               list repair;
               repair[1]=list(intvec(iden0[size(iden0)][i][1],k));
            }
            else
            {
//--- compare against repair, and add the item appropriately
//--- steps of comparison as before
               for(c=1;c<=size(repair);c++)
               {
                  for(d=1;d<=size(repair[c]);d++)
                  {
                     if(defined(opath)) {kill opath;}
                     def opath=imap(re[2][repair[c][d][1]],path);
                     b=0;
                     while(path[1,b+1]==opath[1,b+1])
                     {
                        b++;
                        if((b>ncols(path)-1)||(b>ncols(opath)-1)) break;
                     }
                     b=int(leadcoef(path[1,b]));
                     string tempstr="EList["+string(eval(repair[c][d][2]))
                                     +"]";
                     if(defined(id1)){kill id1;}
                     ideal id1=fetchInTree(re,repair[c][d][1],b,
                               iden0[size(iden0)][i][1],tempstr,iden0,1);
                     kill tempstr;
                     if((size(reduce(EList[k],std(id1)))==0)&&
                        (size(reduce(id1,std(EList[k])))==0))
                     {
                        repair[c][size(repair[c])+1]=intvec(iden0[size(iden0)][i][1],k);
                        break;
                     }
                  }
                  if(d<=size(repair[c]))
                  {
                     break;
                  }
               }
               if(c>size(repair))
               {
                  repair[size(repair)+1]=list(intvec(iden0[size(iden0)][i][1],k));
               }
            }
         }
      }
   }
   if(defined(repair))
   {
//--- there were further components, add them
      for(c=1;c<=size(repair);c++)
      {
         iden[size(iden)+1]=repair[c];
      }
      kill repair;
   }
//--- up to now only Q-irred components - not C-irred components !!!
   return(iden);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=x2+y2+z11;
   list L=resolve(I);
   prepEmbDiv(L);
}
///////////////////////////////////////////////////////////////////////////////
static proc decompEinX(list BO)
"Internal procedure - no help and no example available
"
{
//--- decomposition of exceptional divisor, non-embedded resolution.
//--- even a single exceptional divisor may be Q-reducible when considered
//--- as divisor on the strict transform

//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
   int i,j,k,de,contact;
   intmat interMat;
   list dcE,tmpList,prList,sa,nullList;
   string mpol,compList;
   def R=basering;
   ideal I;
//----------------------------------------------------------------------------
// pass to divisors on V(J) and throw away components already present as
// previous exceptional divisors
//----------------------------------------------------------------------------
   for(i=1;i<=size(BO[4]);i++)
   {
      I=BO[4][i]+BO[2];
      for(j=i+1;j<=size(BO[4]);j++)
      {
         sa=sat(I,BO[4][j]+BO[2]);
         if(sa[2])
         {
           I=sa[1];
         }
      }
//!!! Practical improvement - not yet implemented:
//!!!hier den Input besser aufbereiten (cf. J. Wahl's example)
//!!!I[1]=x(2)^15*y(2)^9+3*x(2)^10*y(2)^6+3*x(2)^5*y(2)^3+x(2)+1;
//!!!I[2]=x(2)^8*y(2)^6+y(0);
//!!!heuristisch die Ordnung so waehlen, dass y(0) im Prinzip eliminiert
//!!!wird.
//-----------------------------------------------------------------------------
// 1) decompose exceptional divisor (over Q)
// 2) check whether there are C-reducible Q-components
// 3) if necessary, find appropriate field extension of Q to decompose
// 4) in each chart collect information in list dcE and export it
//-----------------------------------------------------------------------------
      prList=primdecGTZ(I);
      for(j=1;j<=size(prList);j++)
      {
        tmpList=grad(prList[j][2]);
        de=tmpList[1];
        interMat=tmpList[2];
        mpol=tmpList[3];
        compList=tmpList[4];
        nullList=tmpList[5];
        contact=Kontakt(prList[j][1],BO[2]);
        tmpList=prList[j][2],de,contact,interMat,mpol,compList,nullList;
        prList[j]=tmpList;
      }
      dcE[i]=prList;
   }
   return(dcE);
}
//////////////////////////////////////////////////////////////////////////////
static proc getMinpoly(poly p)
"Internal procedure - no help and no example available
"
{
//---assume that p is a polynomial in 2 variables and irreducible
//---over Q. Computes an irreducible polynomial mp in one variable
//---over Q such that p splits completely over the splitting field of mp
//---returns mp as a string
//---use a variant of the algorithm of S. Gao
  def R=basering;
  int i,j,k,a,b,m,n;
  intvec v;
  string mp="poly p=t-1;";
  list Li=string(1);
  list re=mp,Li,1;

//---check which variables occur in p
  for(i=1;i<=nvars(basering);i++)
  {
    if(p!=subst(p,var(i),0)){v[size(v)+1]=i;}
  }

//---the polynomial is constant
  if(size(v)==1){return(re);}

//---the polynomial depends only on one variable or is homogeneous
//---in 2 variables
  if((size(v)==2)||((size(v)==3)&&(homog(p))))
  {
     if((size(v)==3)&&(homog(p)))
     {
        p=subst(p,var(v[3]),1);
     }
     ring Rhelp=0,var(v[2]),dp;
     poly p=imap(R,p);
     ring Shelp=0,t,dp;
     poly p=fetch(Rhelp,p);
     int de=deg(p);
     p=simplifyMinpoly(p);
     Li=getNumZeros(p);
     short=0;
     mp="poly p="+string(p)+";";
     re=mp,Li,de;
     setring R;
     return(re);
  }
  v=v[2..size(v)];
  if(size(v)>2){ERROR("getMinpoly:input depends on more then 2 variables");}

//---the general case, the polynomial is considered as polynomial in x an y now
  ring T=0,(x,y),lp;
  ideal M,N;
  M[nvars(R)]=0;
  N[nvars(R)]=0;
  M[v[1]]=x;
  N[v[1]]=y;
  M[v[2]]=y;
  N[v[2]]=x;
  map phi=R,M;
  map psi=R,N;
  poly p=phi(p);
  poly q=psi(p);
  ring Thelp=(0,x),y,dp;
  poly p=imap(T,p);
  poly q=imap(T,q);
  n=deg(p);              //---the degree with respect to y
  m=deg(q);              //---the degree with respect to x
  setring T;
  ring A=0,(u(1..m*(n+1)),v(1..(m+1)*n),x,y,t),dp;
  poly f=imap(T,p);
  poly g;
  poly h;
  for(i=0;i<=m-1;i++)
  {
     for(j=0;j<=n;j++)
     {
        g=g+u(i*(n+1)+j+1)*x^i*y^j;
     }
  }
  for(i=0;i<=m;i++)
  {
     for(j=0;j<=n-1;j++)
     {
        h=h+v(i*n+j+1)*x^i*y^j;
     }
  }
  poly L=f*(diff(g,y)-diff(h,x))+h*diff(f,x)-g*diff(f,y);
//---according to the theory f is absolutely irreducible if and only if
//---L(g,h)=0 has no non-trivial solution g,h
//---(g=diff(f,x),h=diff(f,y) is always a solution)
//---therefore we compute a vector space basis of G
//---G={g in Q[x,y],deg_x(g)<m,|exist h, such that L(g,h)=0}
//---dim(G)=a is the number of factors of f in C[x,y]
  matrix M=coef(L,xy);
  ideal J=M[2,1..ncols(M)];
  option(redSB);
  J=std(J);
  option(noredSB);
  poly gred=reduce(g,J);
  ideal G;
  for(i=1;i<=m*(n+1);i++)
  {
    if(gred!=subst(gred,u(i),0))
    {
       G[size(G)+1]=subst(gred,u(i),1);
    }
  }
  for(i=1;i<=n*(m+1);i++)
  {
    if(gred!=subst(gred,v(i),0))
    {
       G[size(G)+1]=subst(gred,v(i),1);
    }
  }
  for(i=1;i<=m*(n+1);i++)
  {
     G=subst(G,u(i),0);
  }
  for(i=1;i<=n*(m+1);i++)
  {
     G=subst(G,v(i),0);
  }
//---the number of factors in C[x,y]
  a=size(G);
  for(i=1;i<=a;i++)
  {
     G[i]=simplify(G[i],1);
  }
  if(a==1)
  {
//---f is absolutely irreducible
    setring R;
    return(re);
  }
//---let g in G be any non-trivial element (g not in <diff(f,x)>)
//---according to the theory f=product over all c in C of the
//---gcd(f,g-c*diff(f,x))
//---let g_1,...,g_a be a basis of G and write
//---g*g_i=sum a_ij*g_j*diff(f,x)  mod f
//---let B=(a_ij) and ch=det(t*unitmat(a)-B) the characteristic
//---polynomial then the number of distinct irreducible factors
//---of gcd(f,g-c*diff(f,x)) in C[x,y] is equal to the multiplicity
//---of c as a root of ch.
//---in our special situation (f is irreducible over Q) ch should
//---be irreducible and the different roots of ch lead to the
//---factors of f, i.e. ch is the minpoly we are looking for

  poly fh=homog(f,t);
//---homogenization is used to obtain a constant matrix using lift
  ideal Gh=homog(G,t);
  int dh,df;
  df=deg(fh);
  for(i=1;i<=a;i++)
  {
    if(deg(Gh[i])>dh){dh=deg(Gh[i]);}
  }
  for(i=1;i<=a;i++)
  {
    Gh[i]=t^(dh-deg(Gh[i]))*Gh[i];
  }
  ideal GF=simplify(diff(fh,x),1)*Gh,fh;
  poly ch;
  matrix LI;
  matrix B[a][a];
  matrix E=unitmat(a);
  poly gran;
  ideal fac;
  for(i=1;i<=a;i++)
  {
     LI=lift(GF,t^(df-1-dh)*Gh[i]*Gh);
     B=LI[1..a,1..a];
     ch=det(t*E-B);
//---irreducibility test
     fac=factorize(ch,1);
     if(deg(fac[1])==a)
     {
        ch=simplifyMinpoly(ch);
        Li=getNumZeros(ch);
        int de=deg(ch);
        short=0;
        mp="poly p="+string(ch)+";";
        re=mp,Li,de;
        setring R;
        return(re);
     }
  }
  ERROR("getMinpoly:not found:please send the example to the authors");
}
//////////////////////////////////////////////////////////////////////////////
static proc getNumZeros(poly p)
"Internal procedure - no help and no example available
"
{
//--- compute numerically (!!!) the zeros of the minimal polynomial
  def R=basering;
  ring S=0,t,dp;
  poly p=imap(R,p);
  def L=laguerre_solve(p,30);
//!!! practical improvement:
//!!! testen ob die Nullstellen signifikant verschieden sind
//!!! und im Notfall Genauigkeit erhoehen
  list re;
  int i;
  for(i=1;i<=size(L);i++)
  {
    re[i]=string(L[i]);
  }
  setring R;
  return(re);
}
//////////////////////////////////////////////////////////////////////////////
static
proc simplifyMinpoly(poly p)
"Internal procedure - no help and no example available
"
{
//--- describe field extension in a simple way
   p=cleardenom(p);
   int n=int(leadcoef(p));
   int d=deg(p);
   int i,k;
   int re=1;
   number s=1;

   list L=primefactors(n);

   for(i=1;i<=size(L[1]);i++)
   {
      k=L[2][i] mod d;
      s=1/number((L[1][i])^(L[2][i] div d));
      if(!k){p=subst(p,t,s*t);}
   }
   p=cleardenom(p);
   n=int(leadcoef(subst(p,t,0)));
   L=primefactors(n);
   for(i=1;i<=size(L[1]);i++)
   {
      k=L[2][i] mod d;
      s=(L[1][i])^(L[2][i] div d);
      if(!k){p=subst(p,t,s*t);}
   }
   p=cleardenom(p);
   return(p);
}
///////////////////////////////////////////////////////////////////////////////
static proc grad(ideal I)
"Internal procedure - no help and no example available
"
{
//--- computes the number of components over C
//--- for a prime ideal of height 1 over Q
  def R=basering;
  int n=nvars(basering);
  string mp="poly p=t-1;";
  string str=string(1);
  list zeroList=string(1);
  int i,j,k,l,d,e,c,mi;
  ideal Istd=std(I);
  intmat interMat;
  d=dim(Istd);
  if(d==-1){return(list(0,0,mp,str,zeroList));}
  if(d!=1){ERROR("ideal is not one-dimensional");}
  ideal Sloc=std(slocus(I));
  if(deg(Sloc[1])>0)
  {
//---This is only to test that in case of singularities we have only
//---one singular point which is a normal crossing
//---consider the different singular points
    ideal M;
    list pr=minAssGTZ(Sloc);
    if(size(pr)>1){ERROR("grad:more then one singular point");}
    for(l=1;l<=size(pr);l++)
    {
       M=std(pr[l]);
       d=vdim(M);
       if(d!=1)
       {
//---now we have to extend the field
          if(defined(S)){kill S;}
          ring S=0,x(1..n),lp;
          ideal M=fetch(R,M);
          ideal I=fetch(R,I);
          ideal jmap;
          map phi=S,maxideal(1);;
          ideal Mstd=std(M);
//---M has to be in general position with respect to lp, i.e.
//---vdim(M)=deg(M[1])
          poly p=Mstd[1];
          e=vdim(Mstd);
          while(e!=deg(p))
          {
             jmap=randomLast(100);
             phi=S,jmap;
             Mstd=std(phi(M));
             p=Mstd[1];
          }
          I=phi(I);
          kill phi;
//---now it is in general position an M[1] defines the field extension
//---Q[x]/M over Q
          ring Shelp=0,t,dp;
          ideal helpmap;
          helpmap[n]=t;
          map psi=S,helpmap;
          poly p=psi(p);
          ring T=(0,t),x(1..n),lp;
          poly p=imap(Shelp,p);
//---we are now in the polynomial ring over the field Q[x]/M
          minpoly=leadcoef(p);
          ideal M=imap(S,Mstd);
          M=M,var(n)-t;
          ideal I=fetch(S,I);
       }
//---we construct a map phi which maps M to maxideal(1)
       option(redSB);
       ideal Mstd=-simplify(std(M),1);
       option(noredSB);
       for(i=1;i<=n;i++)
       {
         Mstd=subst(Mstd,var(i),-var(i));
         M[n-i+1]=Mstd[i];
       }
       M=M[1..n];
//---go to the localization with respect to <x>
       if(d!=1)
       {
          ring Tloc=(0,t),x(1..n),ds;
          poly p=imap(Shelp,p);
          minpoly=leadcoef(p);
          ideal M=fetch(T,M);
          map phi=T,M;
       }
       else
       {
          ring Tloc=0,x(1..n),ds;
          ideal M=fetch(R,M);
          map phi=R,M;
       }
       ideal I=phi(I);
       ideal Istd=std(I);
       mi=mi+milnor(Istd);
       if(mi>l)
       {
          ERROR("grad:divisor is really singular");
       }
       setring R;
    }
  }
  intvec ind=indepSet(Istd,1)[1];
  for(i=1;i<=n;i++){if(ind[i]) break;}
//---the i-th variable is the independent one
  ring Shelp=0,x(1..n),dp;
  ideal I=fetch(R,I);
  if(defined(S)){kill S;}
  if(i==1){ring S=(0,x(1)),x(2..n),lp;}
  if(i==n){ring S=(0,x(n)),x(1..n-1),lp;}
  if((i!=1)&&(i!=n)){ring S=(0,x(i)),(x(1..i-1),x(i+1..n)),lp;}
//---I is zero-dimensional now
  ideal I=imap(Shelp,I);
  ideal Istd=std(I);
  ideal jmap;
  map phi;
  poly p=Istd[1];
  e=vdim(Istd);
  if(e==1)
  {
     setring R;
     str=string(I);
     list resi=1,interMat,mp,str,zeroList;
     return(resi);
  }
//---move I to general position with respect to lp
  if(e!=deg(p))
  {
     jmap=randomLast(5);
     phi=S,jmap;
     Istd=std(phi(I));
     p=Istd[1];
  }
  while(e!=deg(p))
  {
     jmap=randomLast(100);
     phi=S,jmap;
     Istd=std(phi(I));
     p=Istd[1];
  }
  setring Shelp;
  poly p=imap(S,p);
  list Q=getMinpoly(p);
  int de=Q[3];
  mp=Q[1];
//!!!diese Stelle effizienter machen
//!!!minAssGTZ vermeiden durch direkte Betrachtung von
//!!!p und mp und evtl. Quotientenbildung
//!!!bisher nicht zeitkritisch
  string Tesr="ring Tes=(0,t),("+varstr(R)+"),dp;";
  execute(Tesr);
  execute(mp);
  minpoly=leadcoef(p);
  ideal I=fetch(R,I);
  list pr=minAssGTZ(I);
  ideal allgEbene=randomLast(100)[nvars(basering)];
  int minpts=vdim(std(I+allgEbene));
  ideal tempi;
  j=1;
  for(i=1;i<=size(pr);i++)
  {
    tempi=std(pr[i]+allgEbene);
    if(vdim(tempi)<minpts)
    {
      minpts=vdim(tempi);
      j=i;
    }
  }
  tempi=pr[j];
  str=string(tempi);
  kill interMat;
  setring R;
  intmat interMat[de][de]=intersComp(str,mp,Q[2],str,mp,Q[2]);
  list resi=de,interMat,mp,str,Q[2];
  return(resi);
}
////////////////////////////////////////////////////////////////////////////
static proc Kontakt(ideal I, ideal K)
"Internal procedure - no help and no example available
"
{
//---Let K be a prime ideal and I an ideal not contained in K
//---computes a maximalideal M=<x(1)-a1,...,x(n)-an>, ai in a field
//---extension of Q,  containing I+K and an integer a
//---such that in the localization of the polynomial ring with
//---respect to M the ideal I is not contained in K+M^a+1 but in M^a in
  def R=basering;
  int n=nvars(basering);
  int i,j,k,d,e;
  ideal J=std(I+K);
  if(dim(J)==-1){return(0);}
  ideal W;
//---choice of the maximal ideal M
  for(i=1;i<=n;i++)
  {
     W=std(J,var(i));
     d=dim(W);
     if(d==0) break;
  }
  i=1;k=2;
  while((d)&&(i<n))
  {
     W=std(J,var(i)+var(k));
     d=dim(W);
     if(k==n){i++;k=i;}
     if(k<n){k++;}
  }
  while(d)
  {
    W=std(J,randomid(maxideal(1))[1]);
    d=dim(W);
  }
//---now we have a collection om maximalideals and choose one with dim Q[x]/M
//---minimal
  list pr=minAssGTZ(W);
  d=vdim(std(pr[1]));
  k=1;
  for(i=2;i<=size(pr);i++)
  {
    if(d==1) break;
    e=vdim(std(pr[i]));
    if(e<d){k=i;d=e;}
  }
//---M is fixed now
//---if dim Q[x]/M =1 we localize at M
  ideal M=pr[k];
  if(d!=1)
  {
//---now we have to extend the field
     if(defined(S)){kill S;}
     ring S=0,x(1..n),lp;
     ideal M=fetch(R,M);
     ideal I=fetch(R,I);
     ideal K=fetch(R,K);
     ideal jmap;
     map phi=S,maxideal(1);;
     ideal Mstd=std(M);
//---M has to be in general position with respect to lp, i.e.
//---vdim(M)=deg(M[1])
     poly p=Mstd[1];
     e=vdim(Mstd);
     while(e!=deg(p))
     {
        jmap=randomLast(100);
        phi=S,jmap;
        Mstd=std(phi(M));
        p=Mstd[1];
     }
     I=phi(I);
     K=phi(K);
     kill phi;
//---now it is in general position an M[1] defines the field extension
//---Q[x]/M over Q
     ring Shelp=0,t,dp;
     ideal helpmap;
     helpmap[n]=t;
     map psi=S,helpmap;
     poly p=psi(p);
     ring T=(0,t),x(1..n),lp;
     poly p=imap(Shelp,p);
//---we are now in the polynomial ring over the field Q[x]/M
     minpoly=leadcoef(p);
     ideal M=imap(S,Mstd);
     M=M,var(n)-t;
     ideal I=fetch(S,I);
     ideal K=fetch(S,K);
  }
//---we construct a map phi which maps M to maxideal(1)
  option(redSB);
  ideal Mstd=-simplify(std(M),1);
  option(noredSB);
  for(i=1;i<=n;i++)
  {
    Mstd=subst(Mstd,var(i),-var(i));
    M[n-i+1]=Mstd[i];
  }
  M=M[1..n];
//---go to the localization with respect to <x>
  if(d!=1)
  {
     ring Tloc=(0,t),x(1..n),ds;
     poly p=imap(Shelp,p);
     minpoly=leadcoef(p);
     ideal M=fetch(T,M);
     map phi=T,M;
  }
  else
  {
     ring Tloc=0,x(1..n),ds;
     ideal M=fetch(R,M);
     map phi=R,M;
  }
  ideal K=phi(K);
  ideal I=phi(I);
//---compute the order of I in (Q[x]/M)[[x]]/K
  k=1;d=0;
  while(!d)
  {
     k++;
     d=size(reduce(I,std(maxideal(k)+K)));
  }
  setring R;
  return(k-1);
}
example
{"EXAMPLE:";
   echo = 2;
   ring r = 0,(x,y,z),dp;
   ideal I=x4+z4+1;
   ideal K=x+y2+z2;
   Kontakt(I,K);
}
//////////////////////////////////////////////////////////////////////////////
static proc abstractNC(list BO)
"Internal procedure - no help and no example available
"
{
//--- check normal crossing property
//--- used for passing from embedded to non-embedded resolution
//----------------------------------------------------------------------------
// Initialization
//----------------------------------------------------------------------------
  int i,k,j,flag;
  list L;
  ideal J;
  if(dim(std(cent))>0){return(1);}
//----------------------------------------------------------------------------
// check each exceptional divisor on V(J)
//----------------------------------------------------------------------------
  for(i=1;i<=size(BO[4]);i++)
  {
    if(dim(std(BO[2]+BO[4][i]))>0)
    {
//--- really something to do
      J=radical(BO[4][i]+BO[2]);
      if(deg(std(slocus(J))[1])!=0)
      {
        if(!nodes(J))
        {
//--- really singular, not only nodes ==> not normal crossing
          return(0);
        }
      }
      for(k=1;k<=size(L);k++)
      {
//--- run through previously considered divisors
//--- we do not want to bother with the same one twice
         if((size(reduce(J,std(L[k])))==0)&&(size(reduce(L[k],std(J)))==0))
         {
//--- already considered this one
            flag=1;break;
         }
//--- drop previously considered exceptional divisors from the current one
         J=sat(J,L[k])[1];
         if(deg(std(J)[1])==0)
         {
//--- nothing remaining
            flag=1;break;
         }
      }
      if(flag==0)
      {
//--- add exceptional divisor to the list
         L[size(L)+1]=J;
      }
      flag=0;
    }
  }
//---------------------------------------------------------------------------
// check intersection properties between different exceptional divisors
//---------------------------------------------------------------------------
  for(k=1;k<size(L);k++)
  {
     for(i=k+1;i<=size(L);i++)
     {
        if(!nodes(intersect(L[k],L[i])))
        {
//--- divisors Ek and Ei do not meet in a node but in a singularity
//--- which is not allowed to occur ==> not normal crossing
           return(0);
        }
        for(j=i+1;j<=size(L);j++)
        {
           if(deg(std(L[i]+L[j]+L[k])[1])>0)
           {
//--- three divisors meet simultaneously ==> not normal crossing
              return(0);
           }
        }
     }
   }
//--- we reached this point ==> normal crossing
   return(1);
}
//////////////////////////////////////////////////////////////////////////////
static proc nodes(ideal J)
"Internal procedure - no help and no example available
"
{
//--- check whether at most nodes occur as singularities
   ideal K=std(slocus(J));
   if(deg(K[1])==0){return(1);}
   if(dim(K)>0){return(0);}
   if(vdim(K)!=vdim(std(radical(K)))){return(0);}
   return(1);
}
//////////////////////////////////////////////////////////////////////////////

proc intersectionDiv(list re)
"USAGE:   intersectionDiv(L);
@*        L = list of rings
ASSUME:   L is output of resolution of singularities
          (only case of isolated surface singularities)
COMPUTE:  intersection matrix and genera of the exceptional divisors
          (considered as curves on the strict transform)
RETURN:   list l, where
          l[1]: intersection matrix of exceptional divisors
          l[2]: intvec, genera of exceptional divisors
          l[3]: divisorList, encoding the identification of the divisors
EXAMPLE:  example intersectionDiv;  shows an example
"
{
//----------------------------------------------------------------------------
//--- Computes in case of surface singularities (non-embedded resolution):
//--- the intersection of the divisors (on the surface)
//--- assuming that re=resolve(J)
//----------------------------------------------------------------------------
   def R=basering;
//---Test whether we are in the irreducible surface case
   def S=re[2][1];
   setring S;
   BO[2]=BO[2]+BO[1];              // make sure we are living in the smooth W
   if(dim(std(BO[2]))!=2)
   {
     ERROR("The given original object is not a surface");
   }
   if(dim(std(slocus(BO[2])))>0)
   {
     ERROR("The given original object has non-isolated singularities.");
   }
   setring R;
//----------------------------------------------------------------------------
// Compute a non-embedded resolution from the given embedded one by
// dropping redundant trailing blow-ups
//----------------------------------------------------------------------------
   list resu,tmpiden,templist;
   intvec divcomp;
   int i,j,k,offset1,offset2,a,b,c,d,q,found;
//--- compute non-embedded resolution
   list abst=abstractR(re);
   intvec endiv=abst[1];
   intvec deleted=abst[2];
//--- identify the divisors in the various final charts
   list iden=collectDiv(re,deleted)[2];
                                    // list of final divisors
   list iden0=iden;                 // backup copy of iden for later use

   iden=delete(iden,size(iden));    // drop list of endRings from iden
//---------------------------------------------------------------------------
// In iden, only the final charts should be listed, whereas iden0 contains
// everything.
//---------------------------------------------------------------------------
   for(i=1;i<=size(iden);i++)
   {
     k=size(iden[i]);
     tmpiden=iden[i];
     for(j=k;j>0;j--)
     {
        if(!endiv[iden[i][j][1]])
        {
//---not a final chart
           tmpiden=delete(tmpiden,j);
        }
     }
     if(size(tmpiden)==0)
     {
//--- oops, this divisor does not appear in final charts
        iden=delete(iden,i);
        continue;
     }
     else
     {
        iden[i]=tmpiden;
     }
   }
//---------------------------------------------------------------------------
// Even though the exceptional divisors were irreducible in the embedded
// case, they may very well have become reducible after intersection with
// the strict transform of the original object.
// ===> compute a decomposition for each divisor in each of the final charts
//      and change the entries of iden accordingly
// In particular, it is important to keep track of the identification of the
// components of the divisors in each of the charts
//---------------------------------------------------------------------------
   int n=size(iden);
   for(i=1;i<=size(re[2]);i++)
   {
     if(endiv[i])
     {
        def SN=re[2][i];
        setring SN;
        if(defined(dcE)){kill dcE;}
        list dcE=decompEinX(BO);       // decomposition of exceptional divisors
        export(dcE);
        setring R;
        kill SN;
     }
   }
   if(defined(tmpiden)){kill tmpiden;}
   list tmpiden=iden;
   for(i=1;i<=size(iden);i++)
   {
      for(j=size(iden[i]);j>0;j--)
      {
         def SN=re[2][iden[i][j][1]];
         setring SN;
         if(size(dcE[iden[i][j][2]])==1)
         {
           if(dcE[iden[i][j][2]][1][2]==0)
           {
             tmpiden[i]=delete(tmpiden[i],j);
           }
         }
         setring R;
         kill SN;
      }
   }
   for(i=size(tmpiden);i>0;i--)
   {
      if(size(tmpiden[i])==0)
      {
         tmpiden=delete(tmpiden,i);
      }
   }
   iden=tmpiden;
   kill tmpiden;
   list tmpiden;
//--- change entries of iden accordingly
   for(i=1;i<=size(iden);i++)
   {
//--- first set up new entries in iden if necessary - using the first chart
//--- in which we see the respective exceptional divisor
     if(defined(S)){kill S;}
     def S=re[2][iden[i][1][1]];
//--- considering first entry for i-th divisor
     setring S;
     a=size(dcE[iden[i][1][2]]);
     for(j=1;j<=a;j++)
     {
//--- reducible - add to the list considering each component as an exceptional
//--- divisor in its own right
        list tl;
        tl[1]=intvec(iden[i][1][1],iden[i][1][2],j);
        tmpiden[size(tmpiden)+1]=tl;
        kill tl;
     }
//--- now identify the components in the other charts w.r.t. the ones in the
//--- first chart which have already been added to the list
     for(j=2;j<=size(iden[i]);j++)
     {
//--- considering remaining entries for the same original divisor
       if(defined(S2)){kill S2;}
       def S2=re[2][iden[i][j][1]];
       setring S2;
//--- determine common parent of this ring and re[2][iden[i][1][1]]
       if(defined(opath)){kill opath;}
       def opath=imap(S,path);
       b=1;
       while(opath[1,b]==path[1,b])
       {
          b++;
          if((b>ncols(path))||(b>ncols(opath))) break;
       }
       if(defined(li1)){kill li1;}
       list li1;
//--- fetch the components we have considered in re[2][iden[i][1][1]]
//--- via the resolution tree
       for(k=1;k<=a;k++)
       {
          string tempstr="dcE["+string(eval(iden[i][1][2]))+"]["+string(k)+"][1]";
          if(defined(id1)){kill id1;}
          ideal id1=fetchInTree(re,iden[i][1][1],int(leadcoef(path[1,b-1])),
                      iden[i][j][1],tempstr,iden0,1);
          kill tempstr;
          li1[k]=radical(id1);        // for comparison only the geometric
                                      // object matters
          kill id1;
       }
//--- compare the components we have fetched with the components in the
//--- current ring
       for(k=1;k<=size(dcE[iden[i][j][2]]);k++)
       {
          found=0;
          for(b=1;b<=size(li1);b++)
          {
             if((size(reduce(li1[b],std(dcE[iden[i][j][2]][k][1])))==0)&&
               (size(reduce(dcE[iden[i][j][2]][k][1],std(li1[b]+BO[2])))==0))
             {
                li1[b]=ideal(1);
                tmpiden[size(tmpiden)-a+b][size(tmpiden[size(tmpiden)-a+b])+1]=
                       intvec(iden[i][j][1],iden[i][j][2],k);
                found=1;
                break;
             }
          }
          if(!found)
          {
            if(!defined(repair))
            {
               list repair;
               repair[1]=list(intvec(iden[i][j][1],iden[i][j][2],k));
            }
            else
            {
               for(c=1;c<=size(repair);c++)
               {
                  for(d=1;d<=size(repair[c]);d++)
                  {
                     if(defined(opath)) {kill opath;}
                     def opath=imap(re[2][repair[c][d][1]],path);
                     q=0;
                     while(path[1,q+1]==opath[1,q+1])
                     {
                        q++;
                        if((q>ncols(path)-1)||(q>ncols(opath)-1)) break;
                     }
                     q=int(leadcoef(path[1,q]));
                     string tempstr="dcE["+string(eval(repair[c][d][2]))+"]["+string(eval(repair[c][d][3]))+"][1]";
                     if(defined(id1)){kill id1;}
                     ideal id1=fetchInTree(re,repair[c][d][1],q,
                                  iden[i][j][1],tempstr,iden0,1);
                     kill tempstr;
//!!! sind die nicht schon radical?
                     id1=radical(id1);        // for comparison
                                              // only the geometric
                                              // object matters
                     if((size(reduce(dcE[iden[i][j][2]][k][1],std(id1+BO[2])))==0)&&
                        (size(reduce(id1+BO[2],std(dcE[iden[i][j][2]][k][1])))==0))
                     {
                       repair[c][size(repair[c])+1]=intvec(iden[i][j][1],iden[i][j][2],k);
                       break;
                     }
                  }
                  if(d<=size(repair[c]))
                  {
                     break;
                  }
               }
               if(c>size(repair))
               {
                  repair[size(repair)+1]=list(intvec(iden[i][j][1],iden[i][j][2],k));
               }
            }
         }
       }
     }
     if(defined(repair))
     {
       for(c=1;c<=size(repair);c++)
       {
          tmpiden[size(tmpiden)+1]=repair[c];
       }
       kill repair;
     }
   }
   setring R;
   for(i=size(tmpiden);i>0;i--)
   {
      if(size(tmpiden[i])==0)
      {
        tmpiden=delete(tmpiden,i);
        continue;
      }
   }
   iden=tmpiden;                      // store the modified divisor list
   kill tmpiden;                      // and clean up temporary objects
//---------------------------------------------------------------------------
// Now we have decomposed everything into irreducible components over Q,
// but over C there might still be some reducible ones left:
// Determine the number of components over C.
//---------------------------------------------------------------------------
   n=0;
   for(i=1;i<=size(iden);i++)
   {
      if(defined(S)) {kill S;}
      def S=re[2][iden[i][1][1]];
      setring S;
      divcomp[i]=ncols(dcE[iden[i][1][2]][iden[i][1][3]][4]);
          // number of components of the Q-irreducible curve dcE[iden[i][1][2]]
      n=n+divcomp[i];
      setring R;
   }
//---------------------------------------------------------------------------
// set up the entries Inters[i,j] , i!=j, in the intersection matrix:
// we have to compute the intersection of the exceptional divisors (over C)
//    i.e. we have to work in over appropriate algebraic extension of Q.
// (1) plug the intersection matrices of the components of the same Q-irred.
//     divisor into the correct position in the intersection matrix
// (2) for comparison of Ei,k and Ej,l move to a chart where both divisors
//     are present, fetch the components from the very first chart containing
//     the respective divisor and then compare by using intersComp
// (4) put the result into the correct position in the integer matrix Inters
//---------------------------------------------------------------------------
//--- some initialization
   int comPai,comPaj;
   intvec v,w;
   intmat Inters[n][n];
//--- run through all Q-irreducible exceptional divisors
   for(i=1;i<=size(iden);i++)
   {
      if(divcomp[i]>1)
      {
//--- (1) put the intersection matrix for Ei,k with Ei,l into the correct place
         for(k=1;k<=size(iden[i]);k++)
         {
            if(defined(tempmat)){kill tempmat;}
            intmat tempmat=imap(re[2][iden[i][k][1]],dcE)[iden[i][k][2]][iden[i][k][3]][4];
            if(size(ideal(tempmat))!=0)
            {
               Inters[i+offset1..(i+offset1+divcomp[i]-1),
                      i+offset1..(i+offset1+divcomp[i]-1)]=
                       tempmat[1..nrows(tempmat),1..ncols(tempmat)];
               break;
            }
            kill tempmat;
         }
      }
      offset2=offset1+divcomp[i]-1;
//--- set up the components over C of the i-th exceptional divisor
      if(defined(S)){kill S;}
      def S=re[2][iden[i][1][1]];
      setring S;
      if(defined(idlisti)) {kill idlisti;}
      list idlisti;
      idlisti[1]=dcE[iden[i][1][2]][iden[i][1][3]][6];
      export(idlisti);
      setring R;
//--- run through the remaining exceptional divisors and check whether they
//--- have a chart in common with the i-th divisor
      for(j=i+1;j<=size(iden);j++)
      {
         kill templist;
         list templist;
         for(k=1;k<=size(iden[i]);k++)
         {
            intvec tiv2=findInIVList(1,iden[i][k][1],iden[j]);
            if(size(tiv2)!=1)
            {
//--- tiv2[1] is a common chart for the divisors i and j
              tiv2[4..6]=iden[i][k];
              templist[size(templist)+1]=tiv2;
            }
            kill tiv2;
         }
         if(size(templist)==0)
         {
//--- the two (Q-irred) divisors do not appear in any chart simultaneously
            offset2=offset2+divcomp[j]-1;
            j++;
            continue;
         }
         for(k=1;k<=size(templist);k++)
         {
            if(defined(S)) {kill S;}
//--- set up the components over C of the j-th exceptional divisor
            def S=re[2][iden[j][1][1]];
            setring S;
            if(defined(idlistj)) {kill idlistj;}
            list idlistj;
            idlistj[1]=dcE[iden[j][1][2]][iden[j][1][3]][6];
            export(idlistj);
            if(defined(opath)){kill opath;}
            def opath=imap(re[2][templist[k][1]],path);
            comPaj=1;
            while(opath[1,comPaj]==path[1,comPaj])
            {
              comPaj++;
              if((comPaj>ncols(opath))||(comPaj>ncols(path))) break;
            }
            comPaj=int(leadcoef(path[1,comPaj-1]));
            setring R;
            kill S;
            def S=re[2][iden[i][1][1]];
            setring S;
            if(defined(opath)){kill opath;}
            def opath=imap(re[2][templist[k][1]],path);
            comPai=1;
            while(opath[1,comPai]==path[1,comPai])
            {
               comPai++;
               if((comPai>ncols(opath))||(comPai>ncols(path))) break;
            }
            comPai=int(leadcoef(opath[1,comPai-1]));
            setring R;
            kill S;
            def S=re[2][templist[k][1]];
            setring S;
            if(defined(il)) {kill il;}
            if(defined(jl)) {kill jl;}
            if(defined(str1)) {kill str1;}
            if(defined(str2)) {kill str2;}
            string str1="idlisti";
            string str2="idlistj";
            attrib(str1,"algext",imap(re[2][iden[i][1][1]],dcE)[iden[i][1][2]][iden[i][1][3]][5]);
            attrib(str2,"algext",imap(re[2][iden[j][1][1]],dcE)[iden[j][1][2]][iden[j][1][3]][5]);
            list il=fetchInTree(re,iden[i][1][1],comPai,
                                templist[k][1],str1,iden0,1);
            list jl=fetchInTree(re,iden[j][1][1],comPaj,
                                templist[k][1],str2,iden0,1);
            list nulli=imap(re[2][iden[i][1][1]],dcE)[iden[i][1][2]][iden[i][1][3]][7];
            list nullj=imap(re[2][iden[j][1][1]],dcE)[iden[j][1][2]][iden[j][1][3]][7];
            string mpi=imap(re[2][iden[i][1][1]],dcE)[iden[i][1][2]][iden[i][1][3]][5];
            string mpj=imap(re[2][iden[j][1][1]],dcE)[iden[j][1][2]][iden[j][1][3]][5];
            if(defined(tintMat)){kill tintMat;}
            intmat tintMat=intersComp(il[1],mpi,nulli,jl[1],mpj,nullj);
            kill mpi;
            kill mpj;
            kill nulli;
            kill nullj;
            for(a=1;a<=divcomp[i];a++)
            {
               for(b=1;b<=divcomp[j];b++)
               {
                 if(tintMat[a,b]!=0)
                 {
                    Inters[i+offset1+a-1,j+offset2+b-1]=tintMat[a,b];
                    Inters[j+offset2+b-1,i+offset1+a-1]=tintMat[a,b];
                 }
               }
            }
         }
         offset2=offset2+divcomp[j]-1;
      }
      offset1=offset1+divcomp[i]-1;
   }
   Inters=addSelfInter(re,Inters,iden,iden0,endiv);
   intvec GenusIden;

   list tl_genus;
   a=1;
   for(i=1;i<=size(iden);i++)
   {
      tl_genus=genus_E(re,iden0,iden[i][1]);
      for(j=1;j<=tl_genus[2];j++)
      {
        GenusIden[a]=tl_genus[1];
        a++;
      }
   }

   list retlist=Inters,GenusIden,iden,divcomp;
   return(retlist);
}
example
{"EXAMPLE:";
   echo = 2;
   ring r = 0,(x(1..3)),dp(3);
   ideal J=x(3)^5+x(2)^4+x(1)^3+x(1)*x(2)*x(3);
   list re=resolve(J);
   list di=intersectionDiv(re);
   di;

}
//////////////////////////////////////////////////////////////////////////////
static proc intersComp(string str1,
                string mp1,
                list null1,
                string str2,
                string mp2,
                list null2)
"Internal procedure - no help and no example available
"
{
//--- format of input
//--- str1 : ideal (over field extension 1)
//--- mp1  : minpoly of field extension 1
//--- null1: numerical zeros of minpoly
//--- str2 : ideal (over field extension 2)
//--- mp2  : minpoly of field extension 2
//--- null2: numerical zeros of minpoly

//--- determine intersection matrix of the C-components defined by the input

//---------------------------------------------------------------------------
// Initialization
//---------------------------------------------------------------------------
  int ii,jj,same;
  def R=basering;
  intmat InterMat[size(null1)][size(null2)];
  ring ringst=0,(t,s),dp;
//---------------------------------------------------------------------------
// Add new variables s and t and compare the minpolys and ideals
// to find out whether they are identical
//---------------------------------------------------------------------------
  def S=R+ringst;
  setring S;
  if((mp1==mp2)&&(str1==str2))
  {
    same=1;
  }
//--- define first Q-component/C-components, substitute t by s
  string tempstr="ideal id1="+str1+";";
  execute(tempstr);
  execute(mp1);
  id1=subst(id1,t,s);
  poly q=subst(p,t,s);
  kill p;
//--- define second Q-component/C-components
  tempstr="ideal id2="+str2+";";
  execute(tempstr);
  execute(mp2);
//--- do the intersection
  ideal interId=id1+id2+ideal(p)+ideal(q);
  if(same)
  {
    interId=quotient(interId,t-s);
  }
  interId=std(interId);
//--- refine the comparison by passing to each of the numerical zeros
//--- of the two minpolys
  ideal stid=nselect(interId,1..nvars(R));
  ring compl_st=complex,(s,t),dp;
  def stid=imap(S,stid);
  ideal tempid,tempid2;
  for(ii=1;ii<=size(null1);ii++)
  {
    tempstr="number numi="+null1[ii]+";";
    execute(tempstr);
    tempid=subst(stid,s,numi);
    kill numi;
    for(jj=1;jj<=size(null2);jj++)
    {
       tempstr="number numj="+null2[jj]+";";
       execute(tempstr);
       tempid2=subst(tempid,t,numj);
       kill numj;
       if(size(tempid2)==0)
       {
         InterMat[ii,jj]=1;
       }
    }
  }
//--- sanity check; as both Q-components were Q-irreducible,
//--- summation over all entries of a single row must lead to the same
//--- result, no matter which row is chosen
//--- dito for the columns
  int cou,cou1;
  for(ii=1;ii<=ncols(InterMat);ii++)
  {
    cou=0;
    for(jj=1;jj<=nrows(InterMat);jj++)
    {
      cou=cou+InterMat[jj,ii];
    }
    if(ii==1){cou1=cou;}
    if(cou1!=cou){ERROR("intersComp:matrix has wrong entries");}
  }
  for(ii=1;ii<=nrows(InterMat);ii++)
  {
    cou=0;
    for(jj=1;jj<=ncols(InterMat);jj++)
    {
      cou=cou+InterMat[ii,jj];
    }
    if(ii==1){cou1=cou;}
    if(cou1!=cou){ERROR("intersComp:matrix has wrong entries");}
  }
  return(InterMat);
}
/////////////////////////////////////////////////////////////////////////////
static proc addSelfInter(list re,intmat Inters,list iden,list iden0,intvec endiv)
"Internal procedure - no help and no example available
"
{
//---------------------------------------------------------------------------
// Initialization
//---------------------------------------------------------------------------
   def R=basering;
   int i,j,k,l,a,b;
   int n=size(iden);
   intvec v,w;
   list satlist;
   def T=re[2][1];
   setring T;
   poly p;
   p=var(1);  //any linear form will do,
              //but this one is most convenient
   ideal F=ideal(p);
//----------------------------------------------------------------------------
// lift linear form to every end ring, determine the multiplicity of
// the exceptional divisors and store it in Flist
//----------------------------------------------------------------------------
   list templist;
   intvec tiv;
   for(i=1;i<=size(endiv);i++)
   {
     if(endiv[i]==1)
     {
        kill v;
        intvec v;
        a=0;
        if(defined(S)) {kill S;}
        def S=re[2][i];
        setring S;
        map resi=T,BO[5];
        ideal F=resi(F)+BO[2];
        ideal Ftemp=F;
        list Flist;
        if(defined(satlist)){kill satlist;}
        list satlist;
        for(a=1;a<=size(dcE);a++)
        {
           for(b=1;b<=size(dcE[a]);b++)
           {
              Ftemp=sat(Ftemp,dcE[a][b][1])[1];
           }
        }
        F=sat(F,Ftemp)[1];
        Flist[1]=Ftemp;
        Ftemp=1;
        list pr=primdecGTZ(F);
        v[size(pr)]=0;
        for(j=1;j<=size(pr);j++)
        {
          for(a=1;a<=size(dcE);a++)
          {
             if(j==1)
             {
               kill tiv;
               intvec tiv;
               tiv[size(dcE[a])]=0;
               templist[a]=tiv;
               if(v[j]==1)
               {
                 a++;
                 continue;
               }
             }
             if(dcE[a][1][2]==0)
             {
                a++;
                continue;
             }
             for(b=1;b<=size(dcE[a]);b++)
             {
                if((size(reduce(dcE[a][b][1],std(pr[j][2])))==0)&&
                   (size(reduce(pr[j][2],std(dcE[a][b][1])))==0))
                {
                  templist[a][b]=Vielfachheit(pr[j][1],pr[j][2]);
                  v[j]=1;
                  break;
                }
             }
             if((v[j]==1)&&(j>1)) break;
          }
        }
        kill v;
        intvec v;
        Flist[2]=templist;
     }
   }
//-----------------------------------------------------------------------------
// Now set up all the data:
// 1. run through all exceptional divisors in iden and determine the
//    coefficients c_i of the divisor of F.  ===> civ
// 2. determine the intersection locus of F^bar and the Ei and from this data
//    the F^bar.Ei . ===> intF
//-----------------------------------------------------------------------------
   intvec civ;
   intvec intF;
   intF[ncols(Inters)]=0;
   int offset,comPa,ncomp,vd;
   for(i=1;i<=size(iden);i++)
   {
      ncomp=0;
      for(j=1;j<=size(iden[i]);j++)
      {
        if(defined(S)) {kill S;}
        def S=re[2][iden[i][j][1]];
        setring S;
        if((size(civ)<i+offset+1)&&
           (((Flist[2][iden[i][j][2]][iden[i][j][3]])!=0)||(j==size(iden[i]))))
        {
           ncomp=ncols(dcE[iden[i][j][2]][iden[i][j][3]][4]);
           for(k=1;k<=ncomp;k++)
           {
              civ[i+offset+k]=Flist[2][iden[i][j][2]][iden[i][j][3]];
              if(deg(std(slocus(dcE[iden[i][j][2]][iden[i][j][3]][1]))[1])>0)
              {
                   civ[i+offset+k]=civ[i+k];
              }
           }
        }
        if(defined(interId)) {kill interId;}
        ideal interId=dcE[iden[i][j][2]][iden[i][j][3]][1]+Flist[1];
        if(defined(interList)) {kill interList;}
        list interList;
        interList[1]=string(interId);
        interList[2]=ideal(0);
        export(interList);
        if(defined(doneId)) {kill doneId;}
        if(defined(tempId)) {kill tempId;}
        ideal doneId=ideal(1);
        if(defined(dl)) {kill dl;}
        list dl;
        for(k=1;k<j;k++)
        {
           if(defined(St)) {kill St;}
           def St=re[2][iden[i][k][1]];
           setring St;
           if(defined(str)){kill str;}
           string str="interId="+interList[1]+";";
           execute(str);
           if(deg(std(interId)[1])==0)
           {
             setring S;
             k++;
             continue;
           }
           setring S;
           if(defined(opath)) {kill opath;}
           def opath=imap(re[2][iden[i][k][1]],path);
           comPa=1;
           while(opath[1,comPa]==path[1,comPa])
           {
              comPa++;
              if((comPa>ncols(path))||(comPa>ncols(opath))) break;
           }
           comPa=int(leadcoef(path[1,comPa-1]));
           if(defined(str)) {kill str;}
           string str="interList";
           attrib(str,"algext","poly p=t-1;");
           dl=fetchInTree(re,iden[i][k][1],comPa,iden[i][j][1],str,iden0,1);
           if(defined(tempId)){kill tempId;}
           str="ideal tempId="+dl[1]+";";
           execute(str);
           doneId=intersect(doneId,tempId);
           str="interId="+interList[1]+";";
           execute(str);
           interId=sat(interId,doneId)[1];
           interList[1]=string(interId);
        }
        interId=std(interId);
        if(dim(interId)>0)
        {
           "oops, intersection not a set of points";
           ~;
        }
        vd=vdim(interId);
        if(vd>0)
        {
          for(k=i+offset;k<=i+offset+ncomp-1;k++)
          {
             intF[k]=intF[k]+(vd div ncomp);
          }
        }
      }
      offset=size(civ)-i-1;
   }
   if(defined(tiv)){kill tiv;}
   intvec tiv=civ[2..size(civ)];
   civ=tiv;
   kill tiv;
//-----------------------------------------------------------------------------
// Using the F_total= sum c_i Ei + F^bar, the intersection matrix Inters and
// the f^bar.Ei, determine the selfintersection numbers of the Ei from the
// equation F_total.Ei=0 and store it in the diagonal of Inters.
//-----------------------------------------------------------------------------
   intvec diag=Inters*civ+intF;
   for(i=1;i<=size(diag);i++)
   {
      Inters[i,i]=-diag[i] div civ[i];
   }
   return(Inters);
}
//////////////////////////////////////////////////////////////////////////////
static proc invSort(list re, list #)
"Internal procedure - no help and no example available
"
{
  int i,j,k,markier,EZeiger,offset;
  intvec v,e;
  intvec deleted;
  if(size(#)>0)
  {
     deleted=#[1];
  }
  else
  {
    deleted[size(re[2])]=0;
  }
  list LE,HI;
  def R=basering;
//----------------------------------------------------------------------------
// Go through all rings
//----------------------------------------------------------------------------
  for(i=1;i<=size(re[2]);i++)
  {
     if(deleted[i]){i++;continue}
     def S=re[2][i];
     setring S;
//----------------------------------------------------------------------------
// Determine Invariant
//----------------------------------------------------------------------------
     if((size(BO[3])==size(BO[9]))||(size(BO[3])==size(BO[9])+1))
     {
        if(defined(merk2)){kill merk2;}
        intvec merk2;
        EZeiger=0;
        for(j=1;j<=size(BO[9]);j++)
        {
          offset=0;
          if(BO[7][j]==-1)
          {
             BO[7][j]=size(BO[4])-EZeiger;
          }
          for(k=EZeiger+1;(k<=EZeiger+BO[7][j])&&(k<=size(BO[4]));k++)
          {
             if(BO[6][k]==2)
             {
                offset++;
             }
          }
          EZeiger=EZeiger+BO[7][1];
          merk2[3*j-2]=BO[3][j];
          merk2[3*j-1]=BO[9][j]-offset;
          if(size(invSat[2])>j)
          {
            merk2[3*j]=-invSat[2][j];
          }
          else
          {
            if(j<size(BO[9]))
            {
               "!!!!!problem with invSat";~;
            }
          }
        }
        if((size(BO[3])>size(BO[9])))
        {
          merk2[size(merk2)+1]=BO[3][size(BO[3])];
        }
        if((size(merk2)%3)==0)
        {
          intvec tintvec=merk2[1..size(merk2)-1];
          merk2=tintvec;
          kill tintvec;
        }
     }
     else
     {
         ERROR("This situation should not occur, please send the example
                 to the authors.");
     }
//----------------------------------------------------------------------------
// Save invariant describing current center as an object in this ring
// We also store information on the intersection with the center and the
// exceptional divisors
//----------------------------------------------------------------------------
     cent=std(cent);
     kill e;
     intvec e;
     for(j=1;j<=size(BO[4]);j++)
     {
        if(size(reduce(BO[4][j],std(cent+BO[1])))==0)
        {
           e[j]=1;
        }
        else
        {
           e[j]=0;
        }
     }
     if(size(ideal(merk2))==0)
     {
        markier=1;
     }
     if((size(merk2)%3==0)&&(merk2[size(merk2)]==0))
     {
       intvec blabla=merk2[1..size(merk2)-1];
       merk2=blabla;
       kill blabla;
     }
     if(defined(invCenter)){kill invCenter;}
     list invCenter=cent,merk2,e;
     export invCenter;
//----------------------------------------------------------------------------
// Insert it into correct place in the list
//----------------------------------------------------------------------------
     if(i==1)
     {
       if(!markier)
       {
         HI=intvec(merk2[1]+1),intvec(1);
       }
       else
       {
         HI=intvec(778),intvec(1);   // some really large integer
                                     // will be changed at the end!!!
       }
       LE[1]=HI;
       i++;
       setring R;
       kill S;
       continue;
     }
     if(markier==1)
     {
       if(i==2)
       {
         HI=intvec(777),intvec(2);   // same really large integer-1
         LE[2]=HI;
         i++;
         setring R;
         kill S;
         continue;
       }
       else
       {
         if(ncols(path)==2)
         {
           LE[2][2][size(LE[2][2])+1]=i;
           i++;
           setring R;
           kill S;
           continue;
         }
         else
         {
           markier=0;
         }
       }
     }
     j=1;
     def SOld=re[2][int(leadcoef(path[1,ncols(path)]))];
     setring SOld;
     merk2=invCenter[2];
     setring S;
     kill SOld;
     while(merk2<LE[j][1])
     {
        j++;
        if(j>size(LE)) break;
     }
     HI=merk2,intvec(i);
     if(j<=size(LE))
     {
        if(merk2>LE[j][1])
        {
          LE=insert(LE,HI,j-1);
        }
        else
        {
           while((merk2==LE[j][1])&&(size(merk2)<size(LE[j][1])))
           {
              j++;
              if(j>size(LE)) break;
           }
           if(j<=size(LE))
           {
              if((merk2!=LE[j][1])||(size(merk2)!=size(LE[j][1])))
              {
                LE=insert(LE,HI,j-1);
              }
              else
              {
                LE[j][2][size(LE[j][2])+1]=i;
              }
           }
           else
           {
              LE[size(LE)+1]=HI;
           }
        }
     }
     else
     {
        LE[size(LE)+1]=HI;
     }
     setring R;
     kill S;
  }
  if((LE[1][1]==intvec(778)) && (size(LE)>2))
  {
     LE[1][1]=intvec(LE[3][1][1]+2);     // by now we know what 'sufficiently
     LE[2][1]=intvec(LE[3][1][1]+1);     // large' is
  }
  return(LE);
}
example
{"EXAMPLE:";
   echo = 2;
   ring r = 0,(x(1..3)),dp(3);
   ideal J=x(1)^3-x(1)*x(2)^3+x(3)^2;
   list re=resolve(J,1);
   list di=invSort(re);
   di;
}
/////////////////////////////////////////////////////////////////////////////
static proc addToRE(intvec v,int x,list RE)
"Internal procedure - no help and no example available
"
{
//--- auxilliary procedure for collectDiv,
//--- inserting an entry at the correct place
   int i=1;
   while(i<=size(RE))
   {
      if(v==RE[i][1])
      {
         RE[i][2][size(RE[i][2])+1]=x;
         return(RE);
      }
      if(v>RE[i][1])
      {
         list templist=v,intvec(x);
         RE=insert(RE,templist,i-1);
         return(RE);
      }
      i++;
   }
   list templist=v,intvec(x);
   RE=insert(RE,templist,size(RE));
   return(RE);
}
////////////////////////////////////////////////////////////////////////////

proc collectDiv(list re,list #)
"USAGE:   collectDiv(L);
@*        L = list of rings
ASSUME:   L is output of resolution of singularities
COMPUTE:  list representing the identification of the exceptional divisors
          in the various charts
RETURN:   list l, where
          l[1]: intmat, entry k in position i,j implies BO[4][j] of chart i
                is divisor k (if k!=0)
                if k==0, no divisor corresponding to i,j
          l[2]: list ll, where each entry of ll is a list of intvecs
                entry i,j in list ll[k] implies BO[4][j] of chart i
                is divisor k
          l[3]: list L
EXAMPLE:  example collectDiv;   shows an example
"
{
//------------------------------------------------------------------------
// Initialization
//------------------------------------------------------------------------
  int i,j,k,l,m,maxk,maxj,mPa,oPa,interC,pa,ignoreL,iTotal;
  int mLast,oLast=1,1;
  intvec deleted;
//--- sort the rings by the invariant which controlled the last of the
//--- exceptional divisors
  if(size(#)>0)
  {
     deleted=#[1];
  }
  else
  {
    deleted[size(re[2])]=0;
  }
  list LE=invSort(re,deleted);
  list LEtotal=LE;
  intmat M[size(re[2])][size(re[2])];
  intvec invar,tempiv;
  def R=basering;
  list divList;
  list RE,SE;
  intvec myEi,otherEi,tempe;
  int co=2;

  while(size(LE)>0)
  {
//------------------------------------------------------------------------
// Run through the sorted list LE whose entries are lists containing
// the invariant and the numbers of all rings corresponding to it
//------------------------------------------------------------------------
     for(i=co;i<=size(LE);i++)
     {
//--- i==1 in first iteration:
//--- the original ring which did not arise from a blow-up
//--- hence there are no exceptional divisors to be identified there ;

//------------------------------------------------------------------------
// For each fixed value of the invariant, run through all corresponding
// rings
//------------------------------------------------------------------------
        for(l=1;l<=size(LE[i][2]);l++)
        {
           if(defined(S)){kill S;}
           def S=re[2][LE[i][2][l]];
           setring S;
           if(size(BO[4])>maxj){maxj=size(BO[4]);}
//--- all exceptional divisors, except the last one, were previously
//--- identified - hence we can simply inherit the data from the parent ring
           for(j=1;j<size(BO[4]);j++)
           {
              if(deg(std(BO[4][j])[1])>0)
              {
                k=int(leadcoef(path[1,ncols(path)]));
                k=M[k,j];
                if(k==0)
                {
                   RE=addToRE(LE[i][1],LE[i][2][l],RE);
                   ignoreL=1;
                   break;
                }
                M[LE[i][2][l],j]=k;
                tempiv=LE[i][2][l],j;
                divList[k][size(divList[k])+1]=tempiv;
              }
           }
          if(ignoreL){ignoreL=0;l++;continue;}
//----------------------------------------------------------------------------
// In the remaining part of the procedure, the identification of the last
// exceptional divisor takes place.
// Step 1: check whether there is a previously considered ring with the
//         same parent; if this is the case, we can again inherit the data
// Step 1':check whether the parent had a stored center which it then used
//         in this case, we are dealing with an additional component of this
//         divisor: store it in the integer otherComp
// Step 2: if no appropriate ring was found in step 1, we check whether
//         there is a previously considered ring, in the parent of which
//         the center intersects the same exceptional divisors as the center
//         in our parent.
//         if such a ring does not exist: new exceptional divisor
//         if it exists: see below
//----------------------------------------------------------------------------
           if(path[1,ncols(path)-1]==0)
           {
//--- current ring originated from very first blow-up
//--- hence exceptional divisor is the first one
              M[LE[i][2][l],1]=1;
              if(size(divList)>0)
              {
                 divList[1][size(divList[1])+1]=intvec(LE[i][2][l],j);
              }
              else
              {
                 divList[1]=list(intvec(LE[i][2][l],j));
              }
              l++;
              continue;
           }
           if(l==1)
           {
              list TE=addToRE(LE[i][1],1,SE);
              if(size(TE)!=size(SE))
              {
//--- new value of invariant hence new exceptional divisor
                 SE=TE;
                 divList[size(divList)+1]=list(intvec(LE[i][2][l],j));
                 M[LE[i][2][l],j]=size(divList);
              }
              kill TE;
           }
           for(k=1;k<=size(LEtotal);k++)
           {
              if(LE[i][1]==LEtotal[k][1])
              {
                 iTotal=k;
                 break;
              }
           }
//--- Step 1
           k=1;
           while(LEtotal[iTotal][2][k]<LE[i][2][l])
           {
              if(defined(tempPath)){kill tempPath;}
              def tempPath=imap(re[2][LEtotal[iTotal][2][k]],path);
              if(tempPath[1,ncols(tempPath)]==path[1,ncols(path)])
              {
//--- Same parent, hence we inherit our data
                 m=size(imap(re[2][LEtotal[iTotal][2][k]],BO)[4]);
                 m=M[LEtotal[iTotal][2][k],m];
                 if(m==0)
                 {
                    RE=addToRE(LE[i][1],LE[i][2][l],RE);
                    ignoreL=1;
                    break;
                 }
                 M[LE[i][2][l],j]=m;
                 tempiv=LE[i][2][l],j;
                 divList[m][size(divList[m])+1]=tempiv;
                 break;
              }
              k++;
              if(k>size(LEtotal[iTotal][2])) {break;}
           }
           if(ignoreL){ignoreL=0;l++;continue;}
//--- Step 1', if necessary
           if(M[LE[i][2][l],j]==0)
           {
             int savedCent;
             def SPa1=re[2][int(leadcoef(path[1,ncols(path)]))];
                       // parent ring
             setring SPa1;
             if(size(BO)>9)
             {
                if(size(BO[10])>0)
                {
                   savedCent=1;
                }
             }
             if(!savedCent)
             {
                def SPa2=re[2][int(leadcoef(path[1,ncols(path)]))];
                map lMa=SPa2,lastMap;
                       // map leading from grandparent to parent
                list transBO=lMa(BO);
                       // actually we only need BO[10], but this is an
                       // object not a name
                list tempsat;
                if(size(transBO)>9)
                {
//--- there were saved centers
                   while((k<=size(transBO[10])) & (savedCent==0))
                   {
                      tempsat=sat(transBO[10][k][1],BO[4][size(BO[4])]);
                      if(deg(tempsat[1][1])!=0)
                      {
//--- saved center can be seen in this affine chart
                        if((size(reduce(tempsat[1],std(cent)))==0) &&
                           (size(reduce(cent,tempsat[1]))==0))
                        {
//--- this was the saved center which was used
                           savedCent=1;
                        }
                      }
                      k++;
                   }
                }
                kill lMa;          // clean up temporary objects
                kill tempsat;
                kill transBO;
             }
             setring S;         // back to the ring which we want to consider
             if(savedCent==1)
             {
               vector otherComp;
               otherComp[M[int(leadcoef(path[1,ncols(path)])),size(BO[4])-1]]
                        =1;
             }
             kill savedCent;
             if (defined(SPa2)){kill SPa2;}
             kill SPa1;
           }
//--- Step 2, if necessary
           if(M[LE[i][2][l],j]==0)
           {
//--- we are not done after step 1 and 2
              pa=int(leadcoef(path[1,ncols(path)]));  // parent ring
              tempe=imap(re[2][pa],invCenter)[3];     // intersection there
              kill myEi;
              intvec myEi;
              for(k=1;k<=size(tempe);k++)
              {
                if(tempe[k]==1)
                {
//--- center meets this exceptional divisor
                   myEi[size(myEi)+1]=M[pa,k];
                   mLast=k;
                }
              }
//--- ring in which the last divisor we meet is new-born
              mPa=int(leadcoef(path[1,mLast+2]));
              k=1;
              while(LEtotal[iTotal][2][k]<LE[i][2][l])
              {
//--- perform the same preparations for the ring we want to compare with
                 if(defined(tempPath)){kill tempPath;}
                 def tempPath=imap(re[2][LEtotal[iTotal][2][k]],path);
                                                          // its ancestors
                 tempe=imap(re[2][int(leadcoef(tempPath[1,ncols(tempPath)]))],
                         invCenter)[3];                   // its intersections
                 kill otherEi;
                 intvec otherEi;
                 for(m=1;m<=size(tempe);m++)
                 {
                   if(tempe[m]==1)
                   {
//--- its center meets this exceptional divisor
                     otherEi[size(otherEi)+1]
                          =M[int(leadcoef(tempPath[1,ncols(tempPath)])),m];
                     oLast=m;
                   }
                 }
                 if(myEi!=otherEi)
                 {
//--- not the same center because of intersection properties with the
//--- exceptional divisor
                    k++;
                    if(k>size(LEtotal[iTotal][2]))
                    {
                      break;
                    }
                    else
                    {
                      continue;
                    }
                 }
//----------------------------------------------------------------------------
// Current situation:
// 1. the last exceptional divisor could not be identified by simply
//    considering its parent
// 2. it could not be proved to be a new one by considering its intersections
//    with previous exceptional divisors
//----------------------------------------------------------------------------
                 if(defined(bool1)) { kill bool1;}
                 int bool1=
                       compareE(re,LE[i][2][l],LEtotal[iTotal][2][k],divList);
                 if(bool1)
                 {
//--- found some non-empty intersection
                    if(bool1==1)
                    {
//--- it is really the same exceptional divisor
                       m=size(imap(re[2][LEtotal[iTotal][2][k]],BO)[4]);
                       m=M[LEtotal[iTotal][2][k],m];
                       if(m==0)
                       {
                          RE=addToRE(LE[i][1],LE[i][2][l],RE);
                          ignoreL=1;
                          break;
                       }
                       M[LE[i][2][l],j]=m;
                       tempiv=LE[i][2][l],j;
                       divList[m][size(divList[m])+1]=tempiv;
                       break;
                    }
                    else
                    {
                       m=size(imap(re[2][LEtotal[iTotal][2][k]],BO)[4]);
                       m=M[LEtotal[iTotal][2][k],m];
                       if(m!=0)
                       {
                          otherComp[m]=1;
                       }
                    }
                 }
                 k++;
                 if(k>size(LEtotal[iTotal][2]))
                 {
                   break;
                 }
              }
              if(ignoreL){ignoreL=0;l++;continue;}
              if( M[LE[i][2][l],j]==0)
              {
                 divList[size(divList)+1]=list(intvec(LE[i][2][l],j));
                 M[LE[i][2][l],j]=size(divList);
              }
           }
           setring R;
           kill S;
        }
     }
     LE=RE;
     co=1;
     kill RE;
     list RE;
  }
//----------------------------------------------------------------------------
// Add the strict transform to the list of divisors at the last place
// and clean up M
//----------------------------------------------------------------------------
//--- add strict transform
  for(i=1;i<=size(re[2]);i++)
  {
     if(defined(S)){kill S;}
     def S=re[2][i];
     setring S;
     if(size(reduce(cent,std(BO[2])))==0)
     {
        tempiv=i,0;
        RE[size(RE)+1]=tempiv;
     }
     setring R;
  }
  divList[size(divList)+1]=RE;
//--- drop trailing zero-columns of M
  intvec iv0;
  iv0[nrows(M)]=0;
  for(i=ncols(M);i>0;i--)
  {
    if(intvec(M[1..nrows(M),i])!=iv0) break;
  }
  intmat N[nrows(M)][i];
  for(i=1;i<=ncols(N);i++)
  {
    N[1..nrows(M),i]=M[1..nrows(M),i];
  }
  kill M;
  intmat M=N;
  list retlist=cleanUpDiv(re,M,divList);
  return(retlist);
}
example
{"EXAMPLE:";
   echo = 2;
   ring R=0,(x,y,z),dp;
   ideal I=xyz+x4+y4+z4;
//we really need to blow up curves even if the generic point of
//the curve the total transform is n.c.
//this occurs here in r[2][5]
   list re=resolve(I);
   list di=collectDiv(re);
   di[1];
   di[2];
}
//////////////////////////////////////////////////////////////////////////////
static proc cleanUpDiv(list re,intmat M,list divList)
"Internal procedure - no help and no example available
"
{
//--- It may occur that two different entries of invSort coincide on the
//--- first part up to the last entry of the shorter one. In this case
//--- exceptional divisors may appear in both entries of the invSort-list.
//--- To correct this, we now compare the final collection of Divisors
//--- for coinciding ones.
   int i,j,k,a,oPa,mPa,comPa,mdim,odim;
   def R=basering;
   for(i=1;i<=size(divList)-2;i++)
   {
      if(defined(Sm)){kill Sm;}
      def Sm=re[2][divList[i][1][1]];
      setring Sm;
      mPa=int(leadcoef(path[1,ncols(path)]));
      if(defined(SmPa)){kill SmPa;}
      def SmPa=re[2][mPa];
      setring SmPa;
      mdim=dim(std(BO[1]+cent));
      setring Sm;
      if(mPa==1)
      {
//--- very first divisor originates exactly from the first blow-up
//--- there cannot be any mistake here
         i++;
         continue;
      }
      for(j=i+1;j<=size(divList)-1;j++)
      {
         setring Sm;
         for(k=1;k<=size(divList[j]);k++)
         {
            if(size(findInIVList(1,divList[j][k][1],divList[i]))>1)
            {
//--- same divisor cannot appear twice in the same chart
               k=-1;
               break;
            }
         }
         if(k==-1)
         {
            j++;
            if(j>size(divList)-1) break;
            continue;
         }
         if(defined(opath)){kill opath;}
         def opath=imap(re[2][divList[j][1][1]],path);
         oPa=int(leadcoef(opath[1,ncols(opath)]));
         if(defined(SoPa)){kill SoPa;}
         def SoPa=re[2][oPa];
         setring SoPa;
         odim=dim(std(BO[1]+cent));
         setring Sm;
         if(mdim!=odim)
         {
//--- different dimension ==> cannot be same center
            j++;
            if(j>size(divList)-1) break;
            continue;
         }
         comPa=1;
         while(path[1,comPa]==opath[1,comPa])
         {
           comPa++;
           if((comPa>ncols(path))||(comPa>ncols(opath))) break;
         }
         comPa=int(leadcoef(path[1,comPa-1]));
         if(defined(SPa)){kill SPa;}
         def SPa=re[2][mPa];
         setring SPa;
         if(defined(tempIdE)){kill tempIdE;}
         ideal tempIdE=fetchInTree(re,oPa,comPa,mPa,"cent",divList);
         if((size(reduce(cent,std(tempIdE)))!=0)||
            (size(reduce(tempIdE,std(cent)))!=0))
         {
//--- it is not the same divisor!
            j++;
            if(j>size(divList))
            {
               break;
            }
            else
            {
               continue;
            }
         }
         for(k=1;k<=size(divList[j]);k++)
         {
//--- append the entries of the j-th divisor (which is actually also the i-th)
//--- to the i-th divisor
            divList[i][size(divList[i])+1]=divList[j][k];
         }
         divList=delete(divList,j);  //kill obsolete entry from the list
         for(k=1;k<=nrows(M);k++)
         {
            for(a=1;a<=ncols(M);a++)
            {
               if(M[k,a]==j)
               {
//--- j-th divisor is actually the i-th one
                  M[k,a]=i;
               }
               if(M[k,a]>j)
               {
//--- index j was deleted from the list ==> all subsequent indices dropped by
//--- one
                  M[k,a]=M[k,a]-1;
               }
            }
         }
         j--;                        //do not forget to consider new j-th entry
      }
   }
   setring R;
   list retlist=M,divList;
   return(retlist);
}
/////////////////////////////////////////////////////////////////////////////
static proc findTrans(ideal Z, ideal E, list notE, list #)
"Internal procedure - no help and no example available
"
{
//---Auxilliary procedure for fetchInTree!
//---Assume E prime ideal, Z+E eqidimensional,
//---ht(E)+r=ht(Z+E). Compute  P=<p[1],...,p[r]> in Z+E, and polynomial f,
//---such that radical(Z+E)=radical((E+P):f)
   int i,j,d,e;
   ideal Estd=std(E);
//!!! alternative to subsequent line:
//!!!           ideal Zstd=std(radical(Z+E));
   ideal Zstd=std(Z+E);
   ideal J=1;
   if(size(#)>0)
   {
      J=#[1];
   }
   if(deg(Zstd[1])==0){return(list(ideal(1),poly(1)));}
   for(i=1;i<=size(notE);i++)
   {
      notE[i]=std(notE[i]);
   }
   ideal Zred=simplify(reduce(Z,Estd),2);
   if(size(Zred)==0){Z,Estd;~;ERROR("Z is contained in E");}
   ideal P,Q,Qstd;
   Q=Estd;
   attrib(Q,"isSB",1);
   d=dim(Estd);
   e=dim(Zstd);
   for(i=1;i<=size(Zred);i++)
   {
      Qstd=std(Q,Zred[i]);
      if(dim(Qstd)<d)
      {
         d=dim(Qstd);
         P[size(P)+1]=Zred[i];
         Q=Qstd;
         attrib(Q,"isSB",1);
         if(d==e) break;
      }
   }
   list pr=minAssGTZ(E+P);
   list sr=minAssGTZ(J+P);
   i=0;
   Q=1;
   list qr;

   while(i<size(pr))
   {
      i++;
      Qstd=std(pr[i]);
      Zred=simplify(reduce(Zstd,Qstd),2);
      if(size(Zred)==0)
      {
        qr[size(qr)+1]=pr[i];
        pr=delete(pr,i);
        i--;
      }
      else
      {
         Q=intersect(Q,pr[i]);
      }
   }
   i=0;
   while(i<size(sr))
   {
      i++;
      Qstd=std(sr[i]+E);
      Zred=simplify(reduce(Zstd,Qstd),2);
      if((size(Zred)!=0)||(dim(Qstd)!=dim(Zstd)))
      {
        Q=intersect(Q,sr[i]);
      }
   }
   poly f;
   for(i=1;i<=size(Q);i++)
   {
      f=Q[i];
      for(e=1;e<=size(qr);e++)
      {
         if(reduce(f,std(qr[e]))==0){f=0;break;}
      }
      for(j=1;j<=size(notE);j++)
      {
         if(reduce(f,notE[j])==0){f=0; break;}
      }
      if(f!=0) break;
   }
   i=0;
   while(f==0)
   {
      i++;
      f=randomid(Q)[1];
      for(e=1;e<=size(qr);e++)
      {
         if(reduce(f,std(qr[e]))==0){f=0;break;}
      }
      for(j=1;j<=size(notE);j++)
      {
         if(reduce(f,notE[j])==0){f=0; break;}
      }
      if(f!=0) break;
      if(i>20)
      {
         ~;
         ERROR("findTrans:Hier ist was faul");
      }
   }

   list resu=P,f;
   return(resu);
}
/////////////////////////////////////////////////////////////////////////////
static proc compareE(list L, int m, int o, list DivL)
"Internal procedure - no help and no example available
"
{
//----------------------------------------------------------------------------
// We want to compare the divisors BO[4][size(BO[4])] of the rings
// L[2][m] and L[2][o].
// In the initialization step, we collect all necessary data from those
// those rings. In particular, we determine at what point (in the resolution
// history) the branches for L[2][m] and L[2][o] were separated, denoting
// the corresponding ring indices by mPa, oPa and comPa.
//----------------------------------------------------------------------------
   def R=basering;
   int i,j,k,len;

//-- find direct parents and branching point in resolution history
   matrix tpm=imap(L[2][m],path);
   matrix tpo=imap(L[2][o],path);
   int m1,o1=int(leadcoef(tpm[1,ncols(tpm)])),
             int(leadcoef(tpo[1,ncols(tpo)]));
   while((i<ncols(tpo)) && (i<ncols(tpm)))
   {
     if(tpm[1,i+1]!=tpo[1,i+1]) break;
     i++;
   }
   int branchpos=i;
   int comPa=int(leadcoef(tpm[1,branchpos]));  // last common ancestor
//----------------------------------------------------------------------------
// simple checks to save us some work in obvious cases
//----------------------------------------------------------------------------
   if((comPa==m1)||(comPa==o1))
   {
//--- one is in the history of the other ==> they cannot give rise
//---                                        to the same divisor
      return(0);
   }
   def T=L[2][o1];
   setring T;
   int dimCo1=dim(std(cent+BO[1]));
   def S=L[2][m1];
   setring S;
   int dimCm1=dim(std(cent+BO[1]));
   if(dimCm1!=dimCo1)
   {
//--- centers do not have same dimension ==> they cannot give rise
//---                                        to the same divisor
      return(0);
   }
//----------------------------------------------------------------------------
// fetch the center via the tree for comparison
//----------------------------------------------------------------------------
   if(defined(invLocus0)) {kill invLocus0;}
   ideal invLocus0=fetchInTree(L,o1,comPa,m1,"cent",DivL);
             // blow down from L[2][o1] to L[2][comPa] and then up to L[2][m1]
   if(deg(std(invLocus0+invCenter[1]+BO[1])[1])!=0)
   {
     setring R;
     return(int(1));
   }
   if(size(BO)>9)
   {
     for(i=1;i<=size(BO[10]);i++)
     {
       if(deg(std(invLocus0+BO[10][i][1]+BO[1])[1])!=0)
       {
         if(dim(std(BO[10][i][1]+BO[1])) >
              dim(std(invLocus0+BO[10][i][1]+BO[1])))
         {
           ERROR("Internal Error: Please send this example to the authors.");
         }
         setring R;
         return(int(2));
       }
     }
   }
   setring R;
   return(int(0));
//----------------------------------------------------------------------------
// Return-Values:
//        TRUE (=1) if the exceptional divisors coincide,
//        TRUE (=2) if the exceptional divisors originate from different
//                  components of the same center
//        FALSE (=0) otherwise
//----------------------------------------------------------------------------
}
//////////////////////////////////////////////////////////////////////////////

proc fetchInTree(list L,
                 int o1,
                 int comPa,
                 int m1,
                 string idname,
                 list DivL,
                 list #);
"Internal procedure - no help and no example available
"
{
//----------------------------------------------------------------------------
// Initialization and Sanity Checks
//----------------------------------------------------------------------------
   int i,j,k,m,branchPos,inJ,exception;
   string algext;
//--- we need to be in L[2][m1]
   def R=basering;
   ideal test_for_the_same_ring=-77;
   def Sm1=L[2][m1];
   setring Sm1;
   if(!defined(test_for_the_same_ring))
   {
//--- we are not in L[2][m1]
      ERROR("basering has to coincide with L[2][m1]");
   }
   else
   {
//--- we are in L[2][m1]
      kill test_for_the_same_ring;
   }
//--- non-embedded case?
   if(size(#)>0)
   {
      inJ=1;
   }
//--- do parameter values make sense?
   if(comPa<1)
   {
     ERROR("Common Parent should at least be the first ring!");
   }
//--- do we need to pass to an algebraic field extension of Q?
   if(typeof(attrib(idname,"algext"))=="string")
   {
      algext=attrib(idname,"algext");
   }
//--- check wheter comPa is in the history of m1
//--- same test for o1 can be done later on (on the fly)
   if(m1==comPa)
   {
      j=1;
      i=ncols(path)+1;
   }
   else
   {
      for(i=1;i<=ncols(path);i++)
      {
        if(int(leadcoef(path[1,i]))==comPa)
        {
//--- comPa occurs in the history
           j=1;
           break;
        }
      }
   }
   branchPos=i;
   if(j==0)
   {
     ERROR("L[2][comPa] not in history of L[2][m1]!");
   }
//----------------------------------------------------------------------------
// Blow down ideal "idname" from L[2][o1] to L[2][comPa], where the latter
// is assumed to be the common parent of L[2][o1] and L[2][m1]
//----------------------------------------------------------------------------
   if(size(algext)>0)
   {
//--- size(algext)>0: case of algebraic extension of base field
      if(defined(tstr)){kill tstr;}
      string tstr="ring So1=(0,t),("+varstr(L[2][o1])+"),("+ordstr(L[2][o1])+");";
      execute(tstr);
      setring So1;
      execute(algext);
      minpoly=leadcoef(p);
      if(defined(id1)) { kill id1; }
      if(defined(id2)) { kill id2; }
      if(defined(idlist)) { kill idlist; }
      execute("int bool2=defined("+idname+");");
      if(bool2==0)
      {
        execute("list ttlist=imap(L[2][o1],"+idname+");");
      }
      else
      {
        execute("list ttlist="+idname+";");
      }
      kill bool2;
      def BO=imap(L[2][o1],BO);
      def path=imap(L[2][o1],path);
      def lastMap=imap(L[2][o1],lastMap);
      ideal id2=1;
      if(defined(notE)){kill notE;}
      list notE;
      intvec nE;
      list idlist;
      for(i=1;i<=size(ttlist);i++)
      {
         if((i==size(ttlist))&&(typeof(ttlist[i])!="string")) break;
         execute("ideal tid="+ttlist[i]+";");
         idlist[i]=list(tid,ideal(1),nE);
         kill tid;
      }
   }
   else
   {
//--- size(algext)==0: no algebraic extension of base needed
      def So1=L[2][o1];
      setring So1;
      if(defined(id1)) { kill id1; }
      if(defined(id2)) { kill id2; }
      if(defined(idlist)) { kill idlist; }
      execute("ideal id1="+idname+";");
      if(deg(std(id1)[1])==0)
      {
//--- problems with findTrans if id1 is empty set
//!!! todo: also correct in if branch!!!
         setring R;
         return(ideal(1));
      }
     // id1=radical(id1);
      ideal id2=1;
      list idlist;
      if(defined(notE)){kill notE;}
      list notE;
      intvec nE;
      idlist[1]=list(id1,id2,nE);
   }
   if(defined(tli)){kill tli;}
   list tli;
   if(defined(id1)) { kill id1; }
   if(defined(id2)) { kill id2; }
   ideal id1;
   ideal id2;
   if(defined(Etemp)){kill Etemp;}
   ideal Etemp;
   for(m=1;m<=size(idlist);m++)
   {
//!!! Duplicate Block!!! All changes also needed below!!!
//!!! no subprocedure due to large data overhead!!!
//--- run through all ideals to be fetched
      id1=idlist[m][1];
      id2=idlist[m][2];
      nE=idlist[m][3];
      for(i=branchPos-1;i<=size(BO[4]);i++)
      {
//--- run through all relevant exceptional divisors
        if(size(reduce(BO[4][i],std(id1+BO[1])))==0)
        {
//--- V(id1) is contained in except. div. i in this chart
           if(size(reduce(id1,std(BO[4][i])))!=0)
           {
//--- V(id1) does not equal except. div. i of this chart
             Etemp=BO[4][i];
             if(npars(basering)>0)
             {
//--- we are in an algebraic extension of the base field
                if(defined(prtemp)){kill prtemp;}
                list prtemp=minAssGTZ(BO[4][i]);  // C-comp. of except. div.
                j=1;
                if(size(prtemp)>1)
                {
//--- more than 1 component
                  Etemp=ideal(1);
                  for(j=1;j<=size(prtemp);j++)
                  {
//--- find correct component
                     if(size(reduce(prtemp[j],std(id1)))==0)
                     {
                        Etemp=prtemp[j];
                        break;
                     }
                  }
                  if(deg(std(Etemp)[1])==0)
                  {
                    ERROR("fetchInTree:something wrong in field extension");
                  }
                }
                prtemp=delete(prtemp,j); // remove this comp. from list
                while(size(prtemp)>1)
                {
//--- collect all the others into prtemp[1]
                  prtemp[1]=intersect(prtemp[1],prtemp[size(prtemp)]);
                  prtemp=delete(prtemp,size(prtemp));
                }
             }
//--- determine tli[1] and tli[2] such that
//--- V(id1) \cap D(id2) = V(tli[1]) \cap D(tli[2]) \cap BO[4][i]
//--- inside V(BO[1]) (and if necessary inside V(BO[1]+BO[2]))
             if(inJ)
             {
                tli=findTrans(id1+BO[2]+BO[1],Etemp,notE,BO[2]);
             }
             else
             {
                tli=findTrans(id1+BO[1],Etemp,notE);
             }
             if(npars(basering)>0)
             {
//--- in algebraic extension: make sure we stay outside the other components
                if(size(prtemp)>0)
                {
                   for(j=1;j<=ncols(prtemp[1]);j++)
                   {
//--- find the (univariate) generator of prtemp[1] which is the remaining
//--- factor from the factorization over the extension field
                      if(size(reduce(prtemp[1][j],std(id1)))>0)
                      {
                         tli[2]=tli[2]*prtemp[1][j];
                      }
                   }
                }
             }
           }
           else
           {
//--- V(id1) equals except. div. i of this chart
             tli[1]=ideal(0);
             tli[2]=ideal(1);
           }
           id1=tli[1];
           id2=id2*tli[2];
           notE[size(notE)+1]=BO[4][i];
           for(j=1;j<=size(DivL);j++)
           {
              if(inIVList(intvec(o1,i),DivL[j]))
              {
                 nE[size(nE)+1]=j;
                 break;
              }
           }
           if(size(nE)<size(notE))
           {
              ERROR("fetchInTree: divisor not found in divL");
           }
        }
        idlist[m][1]=id1;
        idlist[m][2]=id2;
        idlist[m][3]=nE;
      }
//!!! End of Duplicate Block !!!!
   }
   if(o1>1)
   {
      while(int(leadcoef(path[1,ncols(path)]))>=comPa)
      {
         if((int(leadcoef(path[1,ncols(path)]))>comPa)&&
            (int(leadcoef(path[1,ncols(path)-1]))<comPa))
         {
            ERROR("L[2][comPa] not in history of L[2][o1]!");
         }
         def S=basering;
         if(int(leadcoef(path[1,ncols(path)]))==1)
         {
//--- that's the very first ring!!!
           int und_jetzt_raus;
         }
         if(defined(T)){kill T;}
         if(size(algext)>0)
         {
           if(defined(T0)){kill T0;}
           def T0=L[2][int(leadcoef(path[1,ncols(path)]))];
           if(defined(tstr)){kill tstr;}
           string tstr="ring T=(0,t),("
                      +varstr(L[2][int(leadcoef(path[1,ncols(path)]))])+"),("
                      +ordstr(L[2][int(leadcoef(path[1,ncols(path)]))])+");";
           execute(tstr);
           setring T;
           execute(algext);
           minpoly=leadcoef(p);
           kill tstr;
           def BO=imap(T0,BO);
           if(!defined(und_jetzt_raus))
           {
              def path=imap(T0,path);
              def lastMap=imap(T0,lastMap);
           }
           if(defined(idlist)){kill idlist;}
           list idlist=list(list(ideal(1),ideal(1)));
         }
         else
         {
           def T=L[2][int(leadcoef(path[1,ncols(path)]))];
           setring T;
           if(defined(id1)) { kill id1; }
           if(defined(id2)) { kill id2; }
           if(defined(idlist)){kill idlist;}
           list idlist=list(list(ideal(1),ideal(1)));
         }
         setring S;
         if(defined(phi)) { kill phi; }
         map phi=T,lastMap;
//--- now do the actual blowing down ...
         for(m=1;m<=size(idlist);m++)
         {
//--- ... for each entry of idlist separately
            if(defined(id1)){kill id1;}
            if(defined(id2)){kill id2;}
            ideal id1=idlist[m][1]+BO[1];
            ideal id2=idlist[m][2];
            nE=idlist[m][3];
            if(defined(debug_fetchInTree)>0)
            {
               "Blowing down entry",m,"of idlist:";
               setring S;
               "Abbildung:";phi;
               "before preimage";
               id1;
               id2;
            }
            setring T;
            ideal id1=preimage(S,phi,id1);
            ideal id2=preimage(S,phi,id2);
            if(defined(debug_fetchInTree)>0)
            {
               "after preimage";
               id1;
               id2;
            }
            if(size(id2)==0)
            {
//--- preimage of (principal ideal) id2 was zero, i.e.
//--- generator of previous id2 not in image
              setring S;
//--- it might just be one offending factor ==> factorize
              ideal id2factors=factorize(id2[1])[1];
              int zzz=size(id2factors);
              ideal curfactor;
              setring T;
              id2=ideal(1);
              ideal curfactor;
              for(int mm=1;mm<=zzz;mm++)
              {
//--- blow down each factor separately
                 setring S;
                 curfactor=id2factors[mm];
                 setring T;
                 curfactor=preimage(S,phi,curfactor);
                 if(size(curfactor)>0)
                 {
                    id2[1]=id2[1]*curfactor[1];
                 }
              }
              kill curfactor;
              setring S;
              kill curfactor;
              kill id2factors;
              setring T;
              kill mm;
              kill zzz;
              if(defined(debug_fetchInTree)>0)
              {
                 "corrected id2:";
                 id2;
              }
            }
            idlist[m]=list(id1,id2,nE);
            kill id1,id2;
            setring S;
         }
         setring T;
//--- after blowing down we might again be sitting inside a relevant
//--- exceptional divisor
         for(m=1;m<=size(idlist);m++)
         {
//!!! Duplicate Block!!! All changes also needed above!!!
//!!! no subprocedure due to large data overhead!!!
//--- run through all ideals to be fetched
            if(defined(id1)) {kill id1;}
            if(defined(id2)) {kill id2;}
            if(defined(notE)) {kill notE;}
            if(defined(notE)) {kill notE;}
            list notE;
            ideal id1=idlist[m][1];
            ideal id2=idlist[m][2];
            nE=idlist[m][3];
            for(i=branchPos-1;i<=size(BO[4]);i++)
            {
//--- run through all relevant exceptional divisors
               if(size(reduce(BO[4][i],std(id1)))==0)
               {
//--- V(id1) is contained in except. div. i in this chart
                  if(size(reduce(id1,std(BO[4][i])))!=0)
                  {
//--- V(id1) does not equal except. div. i of this chart
                     if(defined(Etemp)) {kill Etemp;}
                     ideal Etemp=BO[4][i];
                     if(npars(basering)>0)
                     {
//--- we are in an algebraic extension of the base field
                        if(defined(prtemp)){kill prtemp;}
                        list prtemp=minAssGTZ(BO[4][i]); // C-comp.except.div.
                        if(size(prtemp)>1)
                        {
//--- more than 1 component
                           Etemp=ideal(1);
                           for(j=1;j<=size(prtemp);j++)
                           {
//--- find correct component
                               if(size(reduce(prtemp[j],std(id1)))==0)
                               {
                                  Etemp=prtemp[j];
                                  break;
                               }
                           }
                           if(deg(std(Etemp)[1])==0)
                           {
                              ERROR("fetchInTree:something wrong in field extension");
                           }
                        }
                        prtemp=delete(prtemp,j); // remove this comp. from list
                        while(size(prtemp)>1)
                        {
//--- collect all the others into prtemp[1]
                           prtemp[1]=intersect(prtemp[1],prtemp[size(prtemp)]);
                           prtemp=delete(prtemp,size(prtemp));
                        }
                     }
                     if(defined(tli)) {kill tli;}
//--- determine tli[1] and tli[2] such that
//--- V(id1) \cap D(id2) = V(tli[1]) \cap D(tli[2]) \cap BO[4][i]
//--- inside V(BO[1]) (and if necessary inside V(BO[1]+BO[2]))
                     if(inJ)
                     {
                        def tli=findTrans(id1+BO[2]+BO[1],Etemp,notE,BO[2]);
                     }
                     else
                     {
                        def tli=findTrans(id1+BO[1],Etemp,notE);
                     }
                    if(npars(basering)>0)
                    {
//--- in algebraic extension: make sure we stay outside the other components
                      if(size(prtemp)>0)
                      {
                         for(j=1;j<=ncols(prtemp[1]);j++)
                         {
//--- find the (univariate) generator of prtemp[1] which is the remaining
//--- factor from the factorization over the extension field
                           if(size(reduce(prtemp[1][j],std(id1)))>0)
                           {
                              tli[2]=tli[2]*prtemp[1][j];
                           }
                         }
                      }
                    }
                  }
                  else
                  {
                     tli[1]=ideal(0);
                     tli[2]=ideal(1);
                  }
                  id1=tli[1];
                  id2=id2*tli[2];
                  notE[size(notE)+1]=BO[4][i];
                  for(j=1;j<=size(DivL);j++)
                  {
                     if(inIVList(intvec(o1,i),DivL[j]))
                     {
                        nE[size(nE)+1]=j;
                        break;
                     }
                  }
                  if(size(nE)<size(notE))
                  {
                     ERROR("fetchInTree: divisor not found in divL");
                  }
               }
               idlist[m][1]=id1;
               idlist[m][2]=id2;
               idlist[m][3]=nE;
            }
//!!! End of Duplicate Block !!!!
         }
         kill S;
         if(defined(und_jetzt_raus))
         {
           kill und_jetzt_raus;
           break;
         }
      }
      if(defined(debug_fetchInTree)>0)
      {
        "idlist after current blow down step:";
        idlist;
      }
   }
   if(defined(debug_fetchInTree)>0)
   {
      "Blowing down ended";
   }
//----------------------------------------------------------------------------
// Blow up ideal id1 from L[2][comPa] to L[2][m1]. To this end, first
// determine the path to follow and save it in path_togo.
//----------------------------------------------------------------------------
   if(m1==comPa)
   {
//--- no further blow ups needed
      if(size(algext)==0)
      {
//--- no field extension ==> we are done
        return(idlist[1][1]);
      }
      else
      {
//--- field extension ==> we need to encode the result
        list retlist;
        for(m=1;m<=size(idlist);m++)
        {
          retlist[m]=string(idlist[m][1]);
        }
        return(retlist);
      }
   }
//--- we need to blow up
   if(defined(path_m1)) { kill path_m1; }
   matrix path_m1=imap(Sm1,path);
   intvec path_togo;
   for(i=1;i<=ncols(path_m1);i++)
   {
      if(path_m1[1,i]>=comPa)
      {
        path_togo=path_togo,int(leadcoef(path_m1[1,i]));
      }
   }
   path_togo=path_togo[2..size(path_togo)],m1;
   i=1;
   while(i<size(path_togo))
   {
//--- we need to blow up following the path path_togo through the tree
      def S=basering;
      if(defined(T)){kill T;}
      if(size(algext)>0)
      {
//--- in an algebraic extension of the base field
         if(defined(T0)){kill T0;}
         def T0=L[2][path_togo[i+1]];
         if(defined(tstr)){kill tstr;}
         string tstr="ring T=(0,t),(" +varstr(T0)+"),(" +ordstr(T0)+");";
         execute(tstr);
         setring T;
         execute(algext);
         minpoly=leadcoef(p);
         kill tstr;
         def path=imap(T0,path);
         def BO=imap(T0,BO);
         def lastMap=imap(T0,lastMap);
         if(defined(phi)){kill phi;}
         map phi=S,lastMap;
         list idlist=phi(idlist);
         if(defined(debug_fetchInTree)>0)
         {
           "in blowing up (algebraic extension case):";
           phi;
           idlist;
         }
      }
      else
      {
         def T=L[2][path_togo[i+1]];
         setring T;
         if(defined(phi)) { kill phi; }
         map phi=S,lastMap;
         if(defined(idlist)) {kill idlist;}
         list idlist=phi(idlist);
         idlist[1][1]=radical(idlist[1][1]);
         idlist[1][2]=radical(idlist[1][2]);
         if(defined(debug_fetchInTree)>0)
         {
           "in blowing up (case without field extension):";
           phi;
           idlist;
         }
      }
      for(m=1;m<=size(idlist);m++)
      {
//--- get rid of new exceptional divisor
         idlist[m][1]=sat(idlist[m][1]+BO[1],BO[4][size(BO[4])])[1];
         idlist[m][2]=sat(idlist[m][2],BO[4][size(BO[4])])[1];
      }
      if(defined(debug_fetchInTree)>0)
      {
        "after saturation:";
        idlist;
      }
      if((size(algext)==0)&&(deg(std(idlist[1][1])[1])==0))
      {
//--- strict transform empty in this chart, it will stay empty till the end
         setring Sm1;
         return(ideal(1));
      }
      kill S;
      i++;
   }
   if(defined(debug_fetchInTree)>0)
   {
      "End of blowing up steps";
   }
//---------------------------------------------------------------------------
// prepare results for returning them
//---------------------------------------------------------------------------
   ideal E,bla;
   intvec kv;
   list retlist;
   for(m=1;m<=size(idlist);m++)
   {
      for(j=2;j<=size(idlist[m][3]);j++)
      {
         kv=findInIVList(1,path_togo[size(path_togo)],DivL[idlist[m][3][j]]);
         if(kv!=intvec(0))
         {
           E=E+BO[4][kv[2]];
         }
      }
      bla=quotient(idlist[m][1]+E,idlist[m][2]);
      retlist[m]=string(bla);
   }
   if(size(algext)==0)
   {
     return(bla);
   }
   return(retlist);
}
/////////////////////////////////////////////////////////////////////////////
static proc findInIVList(int pos, int val, list ivl)
"Internal procedure - no help and no example available
"
{
//--- find entry with value val at position pos in list of intvecs
//--- and return the corresponding entry
   int i;
   for(i=1;i<=size(ivl);i++)
   {
      if(ivl[i][pos]==val)
      {
         return(ivl[i]);
      }
   }
   return(intvec(0));
}
/////////////////////////////////////////////////////////////////////////////
//static
proc inIVList(intvec iv, list li)
"Internal procedure - no help and no example available
"
{
//--- if intvec iv is contained in list li return 1, 0 otherwise
  int i;
  int s=size(iv);
  for(i=1;i<=size(li);i++)
  {
     if(typeof(li[i])!="intvec"){ERROR("Not integer vector in the list");}
     if(s==size(li[i]))
     {
        if(iv==li[i]){return(1);}
     }
  }
  return(0);
}
//////////////////////////////////////////////////////////////////////////////
static proc Vielfachheit(ideal J,ideal I)
"Internal procedure - no help and no example available
"
{
//--- auxilliary procedure for addSelfInter
//--- compute multiplicity, suitable for the special situation there
  int d=1;
  int vd;
  int c;
  poly p;
  ideal Ip,Jp;
  while((d>0)||(!vd))
  {
     p=randomLast(100)[nvars(basering)];
     Ip=std(I+ideal(p));
     c++;
     if(c>20){ERROR("Vielfachheit: Dimension is wrong");}
     d=dim(Ip);
     vd=vdim(Ip);
  }
  Jp=std(J+ideal(p));
  return(vdim(Jp) div vdim(Ip));
}
/////////////////////////////////////////////////////////////////////////////
static proc genus_E(list re, list iden0, intvec Eindex)
"Internal procedure - no help and no example available
"
{
   int i,ge,gel,num;
   def R=basering;
   ring Rhelp=0,@t,dp;
   def S=re[2][Eindex[1]];
   setring S;
   def Sh=S+Rhelp;
//----------------------------------------------------------------------------
//--- The Q-component X is reducible over C, decomposes into s=num components
//--- X_i, we assume they have n.c.
//---           s*g(X_i)=g(X)+s-1.
//----------------------------------------------------------------------------
   if(defined(I2)){kill I2;}
   ideal I2=dcE[Eindex[2]][Eindex[3]][1];
   num=ncols(dcE[Eindex[2]][Eindex[3]][4]);
   setring Sh;
   if(defined(I2)){kill I2;}
   ideal I2=imap(S,I2);
   I2=homog(I2,@t);
   ge=genus(I2);
   gel=(ge+(num-1)) div num;
   if(gel*num-ge-num+1!=0){ERROR("genus_E: not divisible by num");}
   setring R;
   return(gel,num);
}