singular-data 4.0.3+ds-1 / usr / share / singular / LIB / resjung.lib

////////////////////////////////////////////////////////////////////////////
version="version resjung.lib 4.0.1.0 Nov_2014 "; // $Id: 8a7f52632c2869e127981bdafc45667b1efc800c $
category="Commutative Algebra";
info="
LIBRARY:  jung.lib      Resolution of surface singularities (Desingularization)
                           Algorithm of Jung
AUTHOR:  Philipp Renner,   philipp_renner@web.de

PROCEDURES:
 jungresolve(J[,is_noeth])  computes a resolution (!not a strong one) of the
                            surface given by the ideal J using Jungs Method,
 jungnormal(J[,is_noeth])   computes a representation of J such that all it's
                            singularities are of Hirzebruch-Jung type,
 jungfib(J[,is_noeth])      computes a representation of J such that all it's
                            singularities are quasi-ordinary
";

LIB "resolve.lib";
LIB "mregular.lib";
LIB "sing.lib";
LIB "normal.lib";
LIB "primdec.lib";


//-----------------------------------------------------------------------------------------
//Main procedure
//-----------------------------------------------------------------------------------------

proc jungfib(ideal id, list #)
"USAGE:  jungfib(J[,is_noeth]);
@*       J = ideal
@*       j = int
ASSUME:  J  = two dimensional ideal
RETURN:  a list l of rings
         l[i] is a ring containing two Ideals: QIdeal and BMap.
         BMap defines a birational morphism from V(QIdeal)-->V(J), such that
         V(QIdeal) has only quasi-ordinary singularities.
         If is_noeth=1 the algorithm assumes J is in noether position with respect to
         the last two variables. As a default or if is_noeth = 0 the algorithm computes
         a coordinate change such that J is in noether position.
         NOTE: since the noether position algorithm is randomized the performance
         can vary significantly.
EXAMPLE: example jungfib; shows an example.
"
{
  int noeth = 0;
  if(size(#) == 0)
  {
     #[1]=0;
     noeth=0;
  }
  if(#[1]==1){
    noeth=1;
  }
  ideal I = id;
  I = radical(id);
  def A = basering;
  int n = nvars(A);
  if(deg(NF(1,groebner(slocus(id)))) == -1){
    list result;
    ideal QIdeal = I;
    ideal BMap = maxideal(1);
    export(QIdeal);
    export(BMap);
    result[1] = A;
    return(result);
  }
  if(char(A) <> 0){ERROR("only works for characterisitc 0");}  //dummy check
  if(dim(I)<> 2){ERROR("dimension is unequal 2");}  //dummy check
//Noether Normalization
  if(noeth == 0){
    if(n==3){
      int pos = NoetherP_test(I);
      if(pos ==0){
        ideal noethpos = NoetherPosition(I);
        map phi = A,noethpos;
        kill noethpos,pos;
      }
      else{
        ideal NoetherPos = var(pos);
        for(int i = 1;i<=3;i++){
          if(i<>pos){
            NoetherPos = NoetherPos + var(i);
          }
        }
        map phi = A,NoetherPos;
        kill i,pos,NoetherPos;
      }
    }
    else{
      map phi = A,NoetherPosition(I);
    }
    ideal NoetherN = ideal(phi(I));  //image of id under the NoetherN coordinate change
  }
  else{
    ideal NoetherN = I;
    map phi = A,maxideal(1);
  }
  kill I;
//Critical Locus
  def C2 = branchlocus(NoetherN);
  setring C2;
//dim of critical locus is 0 then the normalization is an resolution
  if(dim(clocus) == 0){
    setring A;
    list nor = normal(NoetherN);
    list result;
    int sizeofnor = size(nor[1]);
    for(int i = 1;i<=sizeofnor;i++){
      def R = nor[1][i];
      setring R;
      ideal QIdeal = norid;
      ideal BMap = BMap;
      export(QIdeal);
      export(BMap);
      result[size(result)+1] = R;
      kill R;
      setring A;
    }
    kill sizeofnor;
    print("This is a resolution.");
    return(result);
  }

//dim of critical locus is 1, so compute embedded resolution of the discriminant curve
  list embresolvee = embresolve(clocus);

//build the fibreproduct
  setring A;
  list fibreP = buildFP(embresolvee,NoetherN,phi);
  //a list of lists, where fibreP[i] contains the information conserning
  //the i-th chart of the fibrepoduct
  //fibreP[i] is the ring; QIdeal the quotientideal; BMap is the map from A
  return(fibreP);
}
example{
 "EXAMPLE:";echo = 2;
//Computing a resolution of singularities of the variety z2-x3-y3
 ring r = 0,(x,y,z),dp;
 ideal I = z2-x3-y3;
//The ideal is in noether position
 list l = jungfib(I,1);
 def R1 = l[1];
 def R2 = l[2];
 setring R1;
 QIdeal;
 BMap;
 setring R2;
 QIdeal;
 BMap;
}

proc jungnormal(ideal id,list #)
"USAGE:  jungnormal(ideal J[,is_noeth]);
@*       J = ideal
@*       i = int
ASSUME:  J  = two dimensional ideal
RETURN:  a list l of rings
         l[i] is a ring containing two Ideals: QIdeal and BMap.
         BMap defines a birational morphism from V(QIdeal)-->V(J), such that
         V(QIdeal) has only singularities of Hizebuch-Jung type.
         If is_noeth=1 the algorithm assumes J is in noether position with respect to
         the last two variables. As a default or if is_noeth = 0 the algorithm computes
         a coordinate change such that J is in noether position.
         NOTE: since the noether position algorithm is randomized the performance
         can vary significantly.
EXAMPLE: example jungnormal; gives an example.
"
{
  int noeth = 0;
  if(size(#) == 0)
  {
     #[1]=0;
     noeth=0;
  }
  if(#[1]==1){
    noeth=1;
  }
  def A = basering;
  list fibreP = jungfib(id,noeth);
  list result;
  for(int i =1;i<=size(fibreP);i++){
    def R1 = fibreP[i];
    setring R1;
    map f1 = A,BMap;
    list nor = normal(QIdeal);
    int sizeofnor = size(nor[1]);
    for(int j = 1;j<=sizeofnor;j++){
      def Ri2 = nor[1][j];
      setring Ri2;
      map f2 = R1,normap;
      ideal BMap = ideal(f2(f1));
      ideal QIdeal = norid;
      export(BMap);
      export(QIdeal);
      result[size(result)+1] = Ri2;
      kill Ri2,f2;
      setring R1;
    }
    kill j,sizeofnor,R1;
  }
  return(result);
}
example{
    "EXAMPLE:";echo = 2;
//Computing a resolution of singularities of the variety z2-x3-y3
    ring r = 0,(x,y,z),dp;
    ideal I = z2-x3-y3;
//The ideal is in noether position
    list l = jungnormal(I,1);
    def R1 = l[1];
    def R2 = l[2];
    setring R1;
    QIdeal;
    BMap;
    setring R2;
    QIdeal;
    BMap;
}

proc jungresolve(ideal id,list #)
"USAGE:  jungresolve(ideal J[,is_noeth]);
@*       J = ideal
@*       i = int
ASSUME:  J  = two dimensional ideal
RETURN:  a list l of rings
         l[i] is a ring containing two Ideals: QIdeal and BMap.
         BMap defines a birational morphism from V(QIdeal)-->V(J), such that
         V(QIdeal) is smooth. For this the algorithm computes first with
         jungnormal a representation of V(J) with Hirzebruch-Jung singularities
         and then it uses Villamayor's algorithm to resolve these singularities
         If is_noeth=1 the algorithm assumes J is in noether position with respect to
         the last two variables. As a default or if is_noeth = 0 the algorithm computes
         a coordinate change such that J is in noether position.
         NOTE: since the noether position algorithm is randomized the performance
         can vary significantly.
EXAMPLE: example jungresolve; shows an example.
"
{
  int noeth = 0;
  if(size(#) == 0)
  {
     #[1]=0;
     noeth=0;
  }
  if(#[1]==1){
    noeth=1;
  }
  def A = basering;
  list result;
  list nor = jungnormal(id,noeth);
  for(int i = 1;i<=size(nor);i++){
    if(defined(R)==voice){kill R;}
    def R3 = nor[i];
    setring R3;
    def R = changeord(list(list("dp",1:nvars(basering))));
    setring R;
    ideal QIdeal = imap(R3,QIdeal);
    ideal BMap = imap(R3,BMap);
    map f = A,BMap;
    if(QIdeal <> 0){
      list res = resolve(QIdeal);
      for(int j =1;j<=size(res[1]);j++){
        def R2 = res[1][j];
        setring R2;
        if(defined(QIdeal)==voice){kill QIdeal;}
        if(defined(BMap)==voice){kill BMap;}
        if(BO[1]<>0){ideal QIdeal = BO[1]+BO[2];}
        else{ideal QIdeal = BO[2];}
        map g = R,BO[5];
        ideal BMap = ideal(g(f));
        export(QIdeal);
        export(BMap);
        result[size(result)+1] = R2;
        kill R2;
      }
      kill j,res;
    }
    else{
      result[size(result)+1] = nor[i];
    }
    setring A;
    kill R,R3;
  }
  return(result);
}
example{
    "EXAMPLE:";echo = 2;
//Computing a resolution of singularities of the variety z2-x3-y3
    ring r = 0,(x,y,z),dp;
    ideal I = z2-x3-y3;
//The ideal is in noether position
    list l = jungresolve(I,1);
    def R1 = l[1];
    def R2 = l[2];
    setring R1;
    QIdeal;
    BMap;
    setring R2;
    QIdeal;
    BMap;
}

//---------------------------------------------------------------------------------------
//Critical locus for the Weierstrass map induced by the noether normalization
//---------------------------------------------------------------------------------------
static proc branchlocus(ideal id)
{
//"USAGE:  branchlocus(ideal J);
//       J = ideal
//ASSUME:  J  = two dimensional ideal in noether position with respect of
//         the last two variables
//RETURN:  A ring containing the ideal clocus respresenting the criticallocus
// of the projection V(J)-->C^2 on the last two coordinates
//EXAMPLE: none"
  def A = basering;
  int n = nvars(A);
  list l = equidim(id);
  int k = size(l);
  ideal LastTwo = var(n-1),var(n);
  ideal lowdim = 1;           //the components of id with dimension smaller 2
  if(k>1){
    for(int j=1;j<k;j++){
      lowdim = intersect(lowdim,radical(l[j]));
    }
  }
  kill k;
  lowdim = radical(lowdim);
  ideal I = radical(l[size(l)]);
  poly product=1;
  kill l;
  for(int i=1; i < n-1; i++){ //elimination of all variables exept var(i),var(n-1),var(n)
    intvec v;
    for(int j=1; j < n-1; j++){
      if(j<>i){
        v[j]=1;
      }
      else{
        v[j]=0;
      }
    }
    v[size(v)+1]=0;
    v[size(v)+1]=0;
    list ringl = ringlist(A);
    list l;
    l[1] = "a";
    l[2] = v;
    list ll = insert(ringl[3],l);
    ringl[3]=ll;
    kill l,ll;
    def R = ring(ringl); //now x_j > x_i > x_n-1 > x_n forall j <> i,n-1,n
    setring R;
    ideal J = groebner(fetch(A,I));//this eliminates the variables
    setring A;
    ideal J = fetch(R,J);
    attrib(J,"isPrincipal",0);
    if(size(J)==1){
      attrib(J,"isPrincipal",1);
    }
    int index = 1;
    if(attrib(J,"isPrincipal")==0){
      setring R;
      for(int j = 1;j<=size(J);j++){//determines the monic polynomial in var(i) with coefficents in C2
        intvec w = leadexp(J[j]);
        attrib(w,"isMonic",1);
        for(int k = 1;k<=size(w);k++){
          if(w[k] <> 0 && k <> i){
            attrib(w,"isMonic",0);
            break;
          }
        }
        //kill k;
        if(attrib(w,"isMonic")==1){
          index = j;
          break;
        }
        kill w;
      }
      kill j;
      setring A;
    }
    product = product*resultant(J[index],diff(J[index],var(i)),var(i));
    //Product of the discriminants, which lies in C2
    kill index,J,v;
  }
  ring C2 = 0,(var(n-1),var(n)),dp;
  setring C2;
  ideal clocus= imap(A,product);      //the critical locus is contained in this
  ideal I = preimage(A,LastTwo,lowdim);
  clocus= radical(intersect(clocus,I));
  //radical is necessary since the resultant is in gerneral not reduced
  export(clocus);
  return(C2);
}

//-----------------------------------------------------------------------------------------
//Build the fibre product of the embedded resolution and the coordinate ring of the variety
//-----------------------------------------------------------------------------------------

static proc buildFP(list embresolve,ideal NoetherN, map phi){
  def A = basering;
  list fibreP;
  int n = nvars(A);
  for(int i=1;i<=size(embresolve);i++){
    def R = embresolve[i];
    setring R;
    list temp = ringlist(A);
    //data for the new ring which is, if A=K[x_1,..,x_n] and
    //R=K[y_1,..,y_m], K[x_1,..,x_n-2,y_1,..,y_m]
    for(int j = 1; j<= nvars(R);j++){
       string st = string(var(j));
       temp[2][n-2+j] = st;
       kill st;
    }
    temp[4] = BO[1];
    ideal J = BO[5];             //ideal of the resolution map
    export(J);
    int m = size(J);
    def R2 = ring(temp);
    kill temp;
    setring R2;
    ideal Temp=0;              //defines map from R to R2 which is the inclusion
    for(int k=n-1;k<n-1+nvars(R);k++){
      Temp = Temp + ideal(var(k));
    }
    map f = R,Temp;
    kill Temp,k;
    ideal FibPMI = ideal(0);    //defines the map from A to R2
    for(int k=1;k<=nvars(A)-m;k++){
      FibPMI=FibPMI+var(k);
    }
    FibPMI= FibPMI+ideal(f(J));
    map FibMap = A,FibPMI;
    kill f,FibPMI;
    ideal TotalT = groebner(FibMap(NoetherN));
    ideal QIdeal = TotalT;
    export(QIdeal);
    ideal FibPMap = ideal(FibMap(phi));
    ideal BMap = FibPMap;
    export(BMap);
    fibreP[i] = R2;
    setring R;
    kill J,R,R2,k,j,m;
  }
  return(fibreP);
}

//-------------------------------------------------------------------------------
//embedded resolution for curves
//-------------------------------------------------------------------------------

static proc embresolve(ideal C)
"USAGE:  embresolve(ideal C);
@*       C = ideal
ASSUME:  C  = ideal of plane curve
RETURN:  a list l of rings
         l[i] is a ring containing a basic object BO, the result of the
         resolution. Whereas the algorithm does not resolve normal
         crossings of V(C)
EXAMPLE: example embresolve shows an example
"
{
  ideal J = 1;
  attrib(J,"iswholeRing",1);
  list primdec = equidim(C);
  if(size(primdec)==2){
  //zero dimensional components of the discrimiant curve are smooth
  //an cross normally so they can be ignored in the resolution process
    ideal Lowdim = radical(primdec[1]);
  }
  else{
    J=radical(C);
  }
  kill primdec;
  list l;
  list BO = createBO(J,l);
  kill J,l;
  list result = resolve2(BO);
  if(defined(Lowdim)==voice)
  {
    for(int i = 1;i<=size(result);i++)
    {
    //had zero dimensional components which I add now to the end result
      def RingforEmbeddedResolution = result[i];
      setring RingforEmbeddedResolution;
      map f = R2,BO[5];
      BO[2]=BO[2]*f(Lowdim);
      kill RingforEmbeddedResolution,f;
    }
  }
  return(result);
}
example
{
  "EXAMPLE:";echo=2;
  //The following curve is the critical locus of the projection z2-x3-y3
  //onto y,z-coordinates.
  ring R = 0,(y,z),dp;
  ideal C = z2-y3;
  list l = embresolve(C);
  def R1 = l[1];
  def R2 = l[2];
  setring R1;
  showBO(BO);
  setring R2;
  showBO(BO);
}

static proc resolve2(list BO){
//computes an embedded resolution for the basic object BO and returns
//a list of rings with BO
  def H = basering;
  setring H;
  attrib(BO[2],"smoothC",0);
  export(BO);
  list result;
  result[1]=H;
  attrib(result[1],"isResolved",0);    //has only simple normal crossings
  attrib(result[1],"smoothC",0);       //has smooth components
  int safety=0;                        //number of runs restricted to 30
  while(1){
    int count2 = 0;         //counts the number of smooth charts
    int p = size(result);
    for(int j = 1;j<=p;j++){
      if(attrib(result[j],"isResolved")==0){
        if(defined(R)){kill R;}
        def R = result[j];
        setring R;
        if(attrib(result[j],"smoothC")==0){
        //has possibly singular components so choose a singular point and blow up
          list primdecPC = primdecGTZ(BO[2]);
          attrib(result[j],"smoothC",1);
          for(int i = 1;i<=size(primdecPC);i++){
            ideal Sl = groebner(slocus(primdecPC[i][2]));
            if(deg(NF(1,Sl))<>-1){
              list primdecSL = primdecGTZ(Sl);
              for(int h =1;h<=size(primdecSL);h++){
                attrib(primdecSL[h],"isRational",1);
              }
              kill h;
              if(!defined(index)){int index = 1;}
              if(defined(blowup)){kill blowup;}
              list blowup = blowUpBO(BO,primdecSL[index][2],3);
              //if it has a rational singularity blow it up else choose
              //some arbitary singular point
              if(attrib(primdecSL[1],"isRational")==0){
              //if we blow up a non rational singularity the exeptional divisors
              //are reduzible so we need to separate them
                for(int k=1;k<=size(blowup);k++){
                  def R2=blowup[k];
                  setring R2;
                  list L;
                  for(int l = 1;l<=size(BO[4]);l++){
                    list primdecED=primdecGTZ(BO[4][l]);
                    L = L + primdecED;
                    kill primdecED;
                  }
                  kill l;
                  BO[4] = L;
                  blowup[k]=R2;
                  kill L,R2;
                }
                kill k;
              }
              kill primdecSL;
              list hlp;
              for(int k = 1;k<j;k++){
                hlp[k]=result[k];
                attrib(hlp[k],"isResolved",attrib(result[k],"isResolved"));
                attrib(hlp[k],"smoothC",attrib(result[k],"smoothC"));
              }
              kill k;
              for(int k =1;k<=size(blowup);k++){
                hlp[size(hlp)+1]=blowup[k];
                attrib(hlp[size(hlp)],"isResolved",0);
                attrib(hlp[size(hlp)],"smoothC",0);
              }
              kill k;
              for(int k = j+1;k<=size(result);k++){
                hlp[size(hlp)+1]=result[k];
                attrib(hlp[size(hlp)],"isResolved",attrib(result[k],"isResolved"));
                attrib(hlp[size(hlp)],"smoothC",attrib(result[k],"smoothC"));
              }
              result = hlp;
              kill hlp,k;
              i=size(primdecPC);
            }
            else{
              attrib(result[j],"smoothC",1);
            }
            kill Sl;
          }
          kill i,primdecPC;
          j=p;
          break;
        }
        else{ //if it has smooth components determine all the intersection
              //points and check whether they are snc or not
          int count = 0;
          ideal Collect = BO[2];
          for(int i = 1;i<=size(BO[4]);i++){
            Collect = Collect*BO[4][i];
          }
          list primdecSL = primdecGTZ(slocus(Collect));
          for(int k = 1;k<=size(primdecSL);k++){
             attrib(primdecSL[k],"isRational",1);

          }
          kill k;
          if(defined(blowup)){kill blowup;}
          list blowup = blowUpBO(BO,primdecSL[1][2],3);
          if(attrib(primdecSL[1],"isRational")==0){
            for(int k=1;k<=size(blowup);k++){
              def R2=blowup[k];
              setring R2;
              list L;
              for(int l = 1;l<=size(BO[4]);l++){
                list primdecED=primdecGTZ(BO[4][l]);
                L = L + primdecED;
                kill primdecED;
              }
              kill l;
              BO[4] = L;
              blowup[k]=R2;
              kill L,R2;
            }
            kill k;
          }
          kill Collect,i;
          for(int i=1;i<=size(primdecSL);i++){
            list L = BO[4];
            L[size(L)+1]=BO[2];
            for(int l = 1;l<=size(L);l++){
              if(L[l][1]==1){L=delete(L,l);}
            }
            kill l;
            if(normalCrossing(ideal(0),L,primdecSL[i][2])==0){
              if(defined(blowup)){kill blowup;}
              list blowup = blowUpBO(BO,primdecSL[i][2],3);
              list hlp;
              for(int k = 1;k<j;k++){
                hlp[k]=result[k];
                attrib(hlp[k],"isResolved",attrib(result[k],"isResolved"));
                attrib(hlp[k],"smoothC",attrib(result[k],"smoothC"));
              }
              kill k;
              for(int k =1;k<=size(blowup);k++){
                hlp[size(hlp)+1]=blowup[k];
                attrib(hlp[size(hlp)],"isResolved",0);
                attrib(hlp[size(hlp)],"smoothC",1);
              }
              kill k;
              for(int k = j+1;k<=size(result);k++){
                hlp[size(hlp)+1]=result[k];
                attrib(hlp[size(hlp)],"isResolved",attrib(result[k],"isResolved"));
                attrib(hlp[size(hlp)],"smoothC",attrib(result[k],"smoothC"));
              }
              result = hlp;
              kill hlp,k;
              j = p;
              break;
            }
            else{
              count++;
            }
            kill L;
          }
          kill i;
          if(count == size(primdecSL)){
            attrib(result[j],"isResolved",1);
          }
          kill count,primdecSL;
        }
        kill R;
      }
      else{
        count2++;
      }
    }
    if(count2==size(result)){
      break;
    }
    kill count2,j,p;
    safety++;
  }
  return(result);
}

static proc NoetherP_test(ideal id)
{
  def A = basering;
  list ringA=ringlist(A);
  int index = 0;
  if(size(id)==1 && nvars(A))
  {  //test if V(id) = C[x,y,z]/<f>
    list L;
    intvec v = 1,1,1;
    L[1] = "lp";
    L[2] = v;
    kill v;
    poly f = id[1];
    int j = 0;
    for(int i = 1;i<=3;i++)
    {
      setring A;
      list l = ringA;  //change ordering to lp and var(i)>var(j) j<>i
      list vari = ringA[2];
      string h = vari[1];
      vari[1] = vari[i];
      vari[i] = h;
      l[2] = vari;
      kill h,vari;
      l[3][1] = L;
      def R = ring(l);
      kill l;
      setring R;
      ideal I = imap(A,id);
      if(defined(v)){kill v;}
      intvec v = leadexp(I[1]);
      attrib(v,"isMonic",1);
      //if(defined(k)==voice){kill k;}
      for(int k = 2;k<=3;k++)
      {  //checks whether f is monic in var(i)
        if(v[k] <> 0 || v[1] == 0)
        {
          attrib(v,"isMonic",0);
          j++;
          break;
        }
      }
      kill k;
      if(attrib(v,"isMonic")==1)
      {
        index = i;
        return(index);
      }
      kill R;
    }
    if(j == 3){ return(0); }
  }
  else{     //not yet a test for more variables
    return(index);
  }
}

////copied from resolve.lib/////////////////
static proc normalCrossing(ideal J,list E,ideal V)
"Internal procedure - no help and no example available
"
{
   int i,d,j;
   int n=nvars(basering);
   list E1,E2;
   ideal K,M,Estd;
   intvec v,w;

   for(i=1;i<=size(E);i++)
   {
      Estd=std(E[i]+J);
      if(deg(Estd[1])>0)
      {
         E1[size(E1)+1]=Estd;
      }
   }
   E=E1;
   for(i=1;i<=size(E);i++)
   {
      v=i;
      E1[i]=list(E[i],v);
   }
   list ll;
   int re=1;

   while((size(E1)>0)&&(re==1))
   {
      K=E1[1][1];
      v=E1[1][2];
      attrib(K,"isSB",1);
      E1=delete(E1,1);
      d=n-dim(K);
      M=minor(jacob(K),d)+K;
      if(deg(std(M+V)[1])>0)
      {
         re=0;
         break;
      }
      for(i=1;i<=size(E);i++)
      {
         for(j=1;j<=size(v);j++){if(v[j]==i){break;}}
         if(j<=size(v)){if(v[j]==i){i++;continue;}}
         Estd=std(K+E[i]);
         w=v;
         if(deg(Estd[1])==0){i++;continue;}
         if(d==n-dim(Estd))
         {
            if(deg(std(Estd+V)[1])>0)
            {
               re=0;
               break;
            }
         }
         w[size(w)+1]=i;
         E2[size(E2)+1]=list(Estd,w);
      }
      if(size(E2)>0)
      {
         if(size(E1)>0)
         {
            E1[size(E1)+1..size(E1)+size(E2)]=E2[1..size(E2)];
         }
         else
         {
            E1=E2;
         }
      }
      kill E2;
      list E2;
   }
   return(re);
}