source: git/Singular/LIB/equising.lib @ 09b9c1

version="$Id: equising.lib,v 1.15 2006-06-12 12:12:27 lossen Exp $";
category="Singularities";
info="
LIBRARY:  equising.lib  Equisingularity Stratum of a Family of Plane Curves
AUTHOR:   Christoph Lossen, lossen@mathematik.uni-kl.de
          Andrea Mindnich, mindnich@mathematik.uni-kl.de

MAIN PROCEDURES:
 tau_es(f);             codim of mu-const stratum in semi-universal def. base
 esIdeal(f);            (Wahl's) equisingularity ideal of f
 esStratum(F[,m,L]);    equisingularity stratum of a family F
 isEquising(F[,m,L]);   tests if a given deformation is equisingular

AUXILIARY PROCEDURE:
 control_Matrix(M);     computes list of blowing-up data
";

LIB "hnoether.lib";
LIB "poly.lib";
LIB "elim.lib";
LIB "deform.lib";
LIB "sing.lib";

////////////////////////////////////////////////////////////////////////////////
//
//  The following (static) procedures are used by esComputation
//
////////////////////////////////////////////////////////////////////////////////
// COMPUTES  a weight vector. x and y get weight 1 and all other
//           variables get weight 0.
static proc xyVector()
{
  intvec iv ;
  iv[nvars(basering)]=0 ;
  iv[rvar(x)] =1;
  iv[rvar(y)] =1;
  return (iv);
}

////////////////////////////////////////////////////////////////////////////////
// exchanges the variables x and y in the polynomial f
static proc swapXY(poly f)
{
  def r_base = basering;
  ideal MI = maxideal(1);
  MI[rvar(x)]=y;
  MI[rvar(y)]=x;
  map phi = r_base, MI;
  f=phi(f);
  return (f);
}

////////////////////////////////////////////////////////////////////////////////
// computes m-jet w.r.t. the variables x,y (other variables weighted 0
static proc m_Jet(poly F,int m);
{
  intvec w=xyVector();
  poly Fd=jet(F,m,w);
  return(Fd);
}


////////////////////////////////////////////////////////////////////////////////
// computes the 4 control matrices (input is multsequence(L))
proc control_Matrix(list M);
"USAGE:   control_Matrix(L); L list
ASSUME:  L is the output of multsequence(hnexpansion(f)).
RETURN:  list M of 4 intmat's:
@format
  M[1] contains the multiplicities at the respective infinitely near points
       p[i,j] (i=step of blowup+1, j=branch) -- if branches j=k,...,k+m pass
       through the same p[i,j] then the multiplicity is stored in M[1][k,j],
       while M[1][k+1]=...=M[1][k+m]=0.
  M[2] contains the number of branches meeting at p[i,j] (again, the information
       is stored according to the above rule)
  M[3] contains the information about the splitting of M[1][i,j] with respect to
       different tangents of branches at p[i,j] (information is stored only for
       minimal j>=k corresponding to a new tangent direction).
       The entries are the sum of multiplicities of all branches with the
       respective tangent.
  M[4] contains the maximal sum of higher multiplicities for a branch passing
       through p[i,j] ( = degree Bound for blowing up)
@end format
NOTE:    the branches are ordered in such a way that only consecutive branches
         can meet at an infinitely near point. @*
         the final rows of the matrices M[1],...,M[3] is (1,1,1,...,1), and
         correspond to infinitely near points such that the strict transforms
         of the branches are smooth and intersect the exceptional divisor
         transversally.
SEE ALSO: multsequence
EXAMPLE: example control_Matrix; shows an example
"
{
  int i,j,k,dummy;

  dummy=0;
  for (j=1;j<=ncols(M[2]);j++)
  {
    dummy=dummy+M[1][nrows(M[1])-1,j]-M[1][nrows(M[1]),j];
  }
  intmat S[nrows(M[1])+dummy][ncols(M[1])];
  intmat T[nrows(M[1])+dummy][ncols(M[1])];
  intmat U[nrows(M[1])+dummy][ncols(M[1])];
  intmat maxDeg[nrows(M[1])+dummy][ncols(M[1])];

  for (i=1;i<=nrows(M[2]);i++)
  {
    dummy=1;
    for (j=1;j<=ncols(M[2]);j++)
    {
      for (k=dummy;k<dummy+M[2][i,j];k++)
      {
        T[i,dummy]=T[i,dummy]+1;
        S[i,dummy]=S[i,dummy]+M[1][i,k];
        if (i>1)
        {
          U[i-1,dummy]=U[i-1,dummy]+M[1][i-1,k];
        }
      }
      dummy=k;
    }
  }

  // adding an extra row (in some cases needed to control ES-Stratum
  // computation)
  for (i=nrows(M[1]);i<=nrows(S);i++)
  {
    for (j=1;j<=ncols(M[2]);j++)
    {
      S[i,j]=1;
      T[i,j]=1;
      U[i,j]=1;
    }
  }

  // Computing the degree Bounds to be stored in M[4]:
  for (i=1;i<=nrows(S);i++)
  {
    dummy=1;
    for (j=1;j<=ncols(S);j++)
    {
      for (k=dummy;k<dummy+T[i,j];k++)
      {
        maxDeg[i,k]=S[i,dummy];  // multiplicity at i-th blowup
      }
      dummy=k;
    }
  }
  // adding up multiplicities
  for (i=nrows(S);i>=2;i--)
  {
    for (j=1;j<=ncols(S);j++)
    {
      maxDeg[i-1,j]=maxDeg[i-1,j]+maxDeg[i,j];
    }
  }

  list L=S,T,U,maxDeg;
  return(L);
}


////////////////////////////////////////////////////////////////////////////////
//  matrix of higher tangent directions:
//  returns list: 1) tangent directions
//                2) swapping information (x <--> y)
static proc inf_Tangents(list L,int s); // L aus hnexpansion,
{
  int nv=nvars(basering);
  matrix M;
  matrix B[s][size(L)];
  intvec V;
  intmat Mult=multsequence(L)[1];

  int i,j,k,counter,e;
  for (k=1;k<=size(L);k++)
  {
    V[k]=L[k][3];  // switch: 0 --> tangent 2nd parameter
                   //         1 --> tangent 1st parameter
    e=0;
    M=L[k][1];
    counter=1;
    B[counter,k]=M[1,1];

    for (i=1;i<=nrows(M);i++)
    {
      for (j=2;j<=ncols(M);j++)
      {
        counter=counter+1;
        if (M[i,j]==var(nv-1))
        {
          if (i<>nrows(M))
          {
            B[counter,k]=M[i,j];
            j=ncols(M)+1; // goto new row of HNmatrix...
            if (counter<>s)
            {
              if (counter+1<=nrows(Mult))
              {
                e=Mult[counter-1,k]-Mult[counter,k]-Mult[counter+1,k];
              }
              else
              {
                e=Mult[counter-1,k]-Mult[counter,k]-1;
              }
            }
          }
          else
          {
            B[counter,k]=0;
            j=ncols(M)+1; // goto new row of HNmatrix...
          }
        }
        else
        {
          if (e<=0)
          {
            B[counter,k]=M[i,j];
          }
          else  // point is still proximate to an earlier point
          {
            B[counter,k]=y; // marking proximity (without swap....)
            if (counter+1<=nrows(Mult))
            {
              e=e-Mult[counter+1,k];
            }
            else
            {
              e=e-1;
            }
          }
        }

        if (counter==s) // given number of points determined
        {
            j=ncols(M)+1;
            i=nrows(M)+1;
            // leave procedure
        }
      }
    }
  }
  L=B,V;
  return(L);
}

////////////////////////////////////////////////////////////////////////////////
// compute "good" upper bound for needed number of help variables
//
static proc Determine_no_b(intmat U,matrix B)
// U is assumed to be 3rd output of control_Matrix
// B is assumed to be 1st output of inf_Tangents
{
  int nv=nvars(basering);
  int i,j,counter;
  for (j=1;j<=ncols(U);j++)
  {
    for (i=1;i<=nrows(U);i++)
    {
      if (U[i,j]>1)
      {
        if (B[i,j]<>var(nv-1) and B[i,j]<>var(nv))
        {
          counter=counter+1;
        }
      }

    }
  }
  counter=counter+ncols(U);
  return(counter);
}

////////////////////////////////////////////////////////////////////////////////
// compute number of infinitely near free points corresponding to non-zero
// entries in control_Matrix[1] (except first row)
//
static proc no_freePoints(intmat Mult,matrix B)
// Mult is assumed to be 1st output of control_Matrix
// U is assumed to be 3rd output of control_Matrix
// B is assumed to be 1st output of inf_Tangents
{
  int i,j,k,counter;
  for (j=1;j<=ncols(Mult);j++)
  {
    for (i=2;i<=nrows(Mult);i++)
    {
      if (Mult[i,j]>=1)
      {
        if (B[i-1,j]<>x and B[i-1,j]<>y)
        {
          counter=counter+1;
        }
      }
    }
  }
  return(counter);
}


///////////////////////////////////////////////////////////////////////////////
// COMPUTES string(minpoly) and substitutes the parameter by newParName
static proc makeMinPolyString (string newParName)
{
  int i;
  string parName = parstr(basering);
  int parNameSize = size(parName);

  string oldMinPolyStr = string (minpoly);
  int minPolySize = size(oldMinPolyStr);

  string newMinPolyStr = "";

  for (i=1;i <= minPolySize; i++)
  {
    if (oldMinPolyStr[i,parNameSize] == parName)
    {
      newMinPolyStr = newMinPolyStr + newParName;
      i = i + parNameSize-1;
    }
    else
    {
      newMinPolyStr = newMinPolyStr + oldMinPolyStr[i];
    }
  }

  return(newMinPolyStr);
}


///////////////////////////////////////////////////////////////////////////////
//
// DEFINES: A new basering, "myRing",
//          with new names for the parameters and variables.
//          The new names for the parameters are a(1..k),
//          and t(1..s),x,y for the variables
//          The ring ordering is ordStr.
// NOTE:    This proc uses 'execute'.
static proc createMyRing_new(poly p_F, string ordStr,
                                string minPolyStr, int no_b)
{
  def r_old = basering;

  int chara = char(basering);
  string charaStr;
  int i;
  string helpStr;
  int nDefParams = nvars(r_old)-2;

  ideal qIdeal = ideal(basering);

  if ((npars(basering)==0) and (minPolyStr==""))
  {
    helpStr = "ring myRing1 ="
              + string(chara)+ ", (t(1..nDefParams), x, y),("+ ordStr +");";
    execute(helpStr);
  }
  else
  {
    charaStr = charstr(basering);
    if (charaStr == string(chara) + "," + parstr(basering) or minPolyStr<>"")
    {
      if (minPolyStr<>"")
      {
        helpStr = "ring myRing1 =
                 (" + string(chara) + ",a),
                 (t(1..nDefParams), x, y),(" + ordStr + ");";
        execute(helpStr);

        execute (minPolyStr);
      }
      else // no minpoly given
      {
        helpStr = "ring myRing1 =
                  (" + string(chara) + ",a(1..npars(basering)) ),
                  (t(1..nDefParams), x, y),(" + ordStr + ");";
        execute(helpStr);
      }
    }
    else
    {
      // ground field is of type (p^k,a)....
      i = find (charaStr,",");
      helpStr = "ring myRing1 = (" + charaStr[1,i] + "a),
              (t(1..nDefParams), x, y),(" + ordStr + ");";
      execute (helpStr);
    }
  }

  ideal mIdeal = maxideal(1);
  ideal qIdeal = fetch(r_old, qIdeal);
  poly p_F = fetch(r_old, p_F);
  export p_F,mIdeal;

  // Extension by no_b auxiliary variables
  if (no_b>0)
  {
    if (npars(basering) == 0)
    {
      ordStr = "(dp("+string(no_b)+"),"+ordStr+")";
      helpStr = "ring myRing ="
                + string(chara)+ ", (b(1..no_b), t(1..nDefParams), x, y),"
                + ordStr +";";
      execute(helpStr);
    }
    else
    {
      charaStr = charstr(basering);
      if (charaStr == string(chara) + "," + parstr(basering))
      {
        if (minpoly !=0)
        {
          ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")";
          minPolyStr = makeMinPolyString("a");
          helpStr = "ring myRing =
                   (" + string(chara) + ",a),
                   (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";";
          execute(helpStr);

          helpStr = "minpoly =" + minPolyStr + ";";
          execute (helpStr);
        }
        else // no minpoly given
        {
          ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")";
          helpStr = "ring myRing =
                    (" + string(chara) + ",a(1..npars(basering)) ),
                    (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";";
          execute(helpStr);
        }
      }
      else
      {
        i = find (charaStr,",");
        ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")";
        helpStr = "ring myRing =
                (" + charaStr[1,i] + "a),
                (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";";
        execute (helpStr);
      }
    }
    ideal qIdeal = imap(myRing1, qIdeal);

    if(qIdeal != 0)
    {
      def r_base = basering;
      setring r_base;
      kill myRing;
      qring myRing = std(qIdeal);
    }

    poly p_F = imap(myRing1, p_F);
    ideal mIdeal = imap(myRing1, mIdeal);
    export p_F,mIdeal;
    kill myRing1;
  }
  else
  {
    if(qIdeal != 0)
    {
      def r_base = basering;
      setring r_base;
      kill myRing1;
      qring myRing = std(qIdeal);
      poly p_F = imap(myRing1, p_F);
      ideal mIdeal = imap(myRing1, mIdeal);
      export p_F,mIdeal;
    }
    else
    {
      def myRing=myRing1;
    }
    kill myRing1;
  }

  setring r_old;
  return(myRing);
}

////////////////////////////////////////////////////////////////////////////////
// returns list of coef, leadmonomial
//
static proc determine_coef (poly Fm)
{
  def r_base = basering; // is assumed to be the result of createMyRing

  int chara = char(basering);
  string charaStr;
  int i;
  string minPolyStr = "";
  string helpStr = "";

  if (npars(basering) == 0)
  {
    helpStr = "ring myRing1 ="
              + string(chara)+ ", (y,x),ds;";
    execute(helpStr);
  }
  else
  {
    charaStr = charstr(basering);
    if (charaStr == string(chara) + "," + parstr(basering))
    {
      if (minpoly !=0)
      {
        minPolyStr = makeMinPolyString("a");
        helpStr = "ring myRing1 = (" + string(chara) + ",a), (y,x),ds;";
        execute(helpStr);

        helpStr = "minpoly =" + minPolyStr + ";";
        execute (helpStr);
      }
      else // no minpoly given
      {
        helpStr = "ring myRing1 =
                  (" + string(chara) + ",a(1..npars(basering)) ), (y,x),ds;";
        execute(helpStr);
      }
    }
    else
    {
      i = find (charaStr,",");

      helpStr = " ring myRing1 = (" + charaStr[1,i] + "a), (y,x),ds;";
      execute (helpStr);
    }
  }
  poly f=imap(r_base,Fm);
  poly g=leadmonom(f);
  setring r_base;
  poly g=imap(myRing1,g);
  kill myRing1;
  def M=coef(Fm,xy);

  for (i=1; i<=ncols(M); i++)
  {
    if (M[1,i]==g)
    {
      poly h=M[2,i];  // determine coefficient of leading monomial (in K[t])
      i=ncols(M)+1;
    }
  }
  return(list(h,g));
}

///////////////////////////////////////////////////////////////////////////////
// RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f
//             or p_f is not squarefree.
//          0, otherwise
static proc checkPoly (poly p_f)
{
  int i_print = printlevel - voice + 3;
  int i_ord;

  if (p_f == 0)
  {
    print("Input is a 'deformation'  of the zero polynomial!");
    return(1);
  }

  i_ord = mindeg1(p_f);

  if (number(i_ord) == 0)
  {
    print("Characteristic of coefficient field "
            +"divides order of zero-fiber !");
    return(1);
  }

  if (squarefree(p_f) != p_f)
  {
    print("Original polynomial (= zero-fiber) is not reduced!");
    return(1);
  }

  return(0);
}

////////////////////////////////////////////////////////////////////////////////
static proc make_ring_small(ideal J)
// returns varstr for new ring, the map and the number of vars
{
  attrib(J,"isSB",1);
  int counter=0;
  ideal newmap;
  string newvar="";
  for (int i=1; i<=nvars(basering); i++)
  {
    if (reduce(var(i),J)<>0)
    {
      newmap[i]=var(i);

      if (newvar=="")
      {
        newvar=newvar+string(var(i));
        counter=counter+1;
      }
      else
      {
        newvar=newvar+","+string(var(i));
        counter=counter+1;
      }
    }
    else
    {
      newmap[i]=0;
    }
  }
  list L=newvar,newmap,counter;
  attrib(J,"isSB",0);
  return(L);
}

///////////////////////////////////////////////////////////////////////////////
//  The following procedure is called by esStratum (typ=0), resp. by
//  isEquising (typ=1)
///////////////////////////////////////////////////////////////////////////////

static proc esComputation (int typ, poly p_F, list #)
{
  intvec ov=option(get);  // store options set at beginning
  option(redSB);
  // Initialize variables
  int branch=1;
  int blowup=1;
  int auxVar=1;
  int nVars;

  intvec upper_bound, upper_bound_old, fertig, soll;
  list blowup_string;
  int i_print= printlevel-voice+2;

  int no_b, number_of_branches, swapped;
  int i,j,k,m, counter, dummy;
  string helpStr = "";
  string ordStr = "";
  string MinPolyStr = "";

  if (nvars(basering)<=2)
  {
    print("family is trivial (no deformation parameters)!");
    if (typ==1) //isEquising
    {
      option(set,ov);
      return(1);
    }
    else
    {
      option(set,ov);
      return(list(ideal(0),0));
    }
  }

  if (size(#)>0)
  {
    if (typeof(#[1])=="int")
    {
      def artin_bd=#[1];  // compute modulo maxideal(artin_bd)
      if (artin_bd <= 1)
      {
        print("Do you really want to compute over Basering/maxideal("
              +string(artin_bd)+") ?");
        print("No computation performed !");
        if (typ==1) //isEquising
        {
          option(set,ov);
          return(1);
        }
        else
        {
          option(set,ov);
          return(list(ideal(0),int(1)));
        }
      }
      if (size(#)>1)
      {
        if (typeof(#[2])=="list")
        {
          def @L=#[2];  // is assumed to be the Hamburger-Noether matrix
        }
      }
    }
    else
    {
      if (typeof(#)=="list")
      {
        def @L=#;  // is assumed to be the Hamburger-Noether matrix
      }
    }
  }
  int ring_is_changed;
  def old_ring=basering;
  if(defined(@L)<=0)
  {
    // define a new ring without deformation-parameters and change to it:
    string str;
    string minpolyStr = string(minpoly);
    str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;";
    execute (str);
    str = "minpoly ="+ minpolyStr+";";
    execute(str);
    ring_is_changed=1;
    // Basering changed to HNERing (variables x,y, with ls ordering)

    k=nvars(old_ring);
    matrix Map_Phi[1][k];
    Map_Phi[1,k-1]=x;
    Map_Phi[1,k]=y;
    map phi=old_ring,Map_Phi;
    poly f=phi(p_F);

    // Heuristics: if x,y are transversal parameters then computation of HNE
    // can be much faster when exchanging variables...!
    if (2*size(coeffs(f,x))<size(coeffs(f,y)))
    {
      swapped=1;
      f=swapXY(f);
    }

    int error=checkPoly(f);
    if (error)
    {
      setring old_ring;
      if (typ==1) //isEquising
      {
        print("Return value (=0) has no meaning!");
        option(set,ov);
        return(0);
      }
      else
      {
        option(set,ov);
        return(list( ideal(0),error));
      }
    }

    dbprint(i_print,"// ");
    dbprint(i_print,"// Compute HN expansion");
    dbprint(i_print,"// ---------------------");
    i=printlevel;
    printlevel=printlevel-5;
    list LLL=hnexpansion(f);

    if (size(LLL)==0) { // empty list returned by hnexpansion
      setring old_ring;
      print(i_print,"Unable to compute HN expansion !");
      if (typ==1) //isEquising
      {
        print("Return value (=0) has no meaning!");
        option(set,ov);
        return(0);
      }
      else
      {
        option(set,ov);
        return(list(ideal(0),int(1)));
      }
      option(set,ov);
      return(0);
    }
    else
    {
      if (typeof(LLL[1])=="ring") {
        def HNering = LLL[1];
        setring HNering;
        def @L=stripHNE(hne);
      }
      else {
        def @L=stripHNE(LLL);
      }
    }
    printlevel=i;
    dbprint(i_print,"// finished");
    dbprint(i_print,"// ");
  }
  def HNEring=basering;
  list M=multsequence(@L);
  M=control_Matrix(M);     // this returns the 4 control matrices
  def maxDeg=M[4];

  list L1=inf_Tangents(@L,nrows(M[1]));
  matrix B=L1[1];
  intvec V=L1[2];
  kill L1;

  // if we have computed the HNE for f after swapping x and y, we have
  // to reinterprete the (swap) matrix V:
  if (swapped==1)
  {
    for (i=1;i<=size(V);i++) { V[i]=V[i]-1; } // turns 0 into -1, 1 into 0
  }

  // Determine maximal number of needed auxiliary parameters (free tangents):
  no_b=Determine_no_b(M[3],B);

  // test whether HNexpansion needed field extension....
  string minPolyStr = "";
  if (minpoly !=0)
  {
    minPolyStr = makeMinPolyString("a");
    minPolyStr = "minpoly =" + minPolyStr + ";";
  }

  setring old_ring;

  def myRing=createMyRing_new(p_F,"dp",minPolyStr,no_b);
  setring myRing;  // comes with mIdeal
  map hole=HNEring,mIdeal;
  // basering has changed to myRing, in particular, the "old"
  // variable names, e.g., A,B,C,z,y are replaced by t(1),t(2),t(3),x,y

  ideal bNodes;

  // Initialize some variables:
  map phi;
  poly G, F_save;
  poly b_dummy;
  ideal J,Jnew,final_Map;
  number_of_branches=ncols(M[1]);
  for (i=1;i<=number_of_branches;i++)
  {
    poly F(i);
    ideal bl_Map(i);
  }
  upper_bound[number_of_branches]=0;
  upper_bound[1]=number_of_branches;
  upper_bound_old=upper_bound;
  fertig[number_of_branches]=0;
  for (i=1;i<=number_of_branches;i++){ soll[i]=1; }

  // Hole:  B = matrix of blowup points
  if (ring_is_changed==0) { matrix B=hole(B); }
  else                    { matrix B=imap(HNEring,B); }
  m=M[1][blowup,branch];    // multiplicity at 0

  // now, we start by checking equimultiplicity along trivial section
  poly Fm=m_Jet(p_F,m-1);

  matrix coef_Mat = coef(Fm,xy);
  Jnew=coef_Mat[2,1..ncols(coef_Mat)];
  J=J,Jnew;

  if (defined(artin_bd)) // the artin_bd-th power of the maxideal of
                         // deformation parameters can be cutted off
  {
    J=jet(J,artin_bd-1);
  }

  J=interred(J);
  if (defined(artin_bd)) { J=jet(J,artin_bd-1); }

// J=std(J);

  if (typ==1) // isEquising
  {
    if(ideal(nselect(J,1,no_b))<>0)
    {
      setring old_ring;
      option(set,ov);
      return(0);
    }
  }

  F(1)=p_F;

  // and reduce the remaining terms in F(1):
  bl_Map(1)=maxideal(1);

  attrib(J,"isSB",1);
  bl_Map(1)=reduce(bl_Map(1),J);
  attrib(J,"isSB",0);

  phi=myRing,bl_Map(1);
  F(1)=phi(F(1));

  // simplify F(1)
  attrib(J,"isSB",1);
  F(1)=reduce(F(1),J);
  attrib(J,"isSB",0);

  // now we compute the m-jet:
  Fm=m_Jet(F(1),m);

  G=1;
  counter=branch;
  k=upper_bound[branch];

  F_save=F(1);  // is truncated differently in the following loop

  while(counter<=k)
  {
    F(counter)=m_Jet(F_save,maxDeg[blowup,counter]);
    if (V[counter]==0) // 2nd ring variable is tangent to this branch
    {
      G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]);
    }
    else // 1st ring variable is tangent to this branch
    {
      G=G*(x-(b(auxVar)+B[blowup,counter])*y)^(M[3][blowup,counter]);
      F(counter)=swapXY(F(counter));
    }
    bl_Map(counter)=maxideal(1);
    bl_Map(counter)[nvars(basering)]=xy+(b(auxVar)+B[blowup,counter])*x;

    bNodes[counter]=b(auxVar);

    auxVar=auxVar+1;
    upper_bound[counter]=counter+M[2][blowup+1,counter]-1;
    counter=counter+M[2][blowup+1,counter];

  }

  list LeadDataFm=determine_coef(Fm);
  def LeadDataG=coef(G,xy);

  for (i=1; i<=ncols(LeadDataG); i++)
  {
    if (LeadDataG[1,i]==LeadDataFm[2])
    {
      poly LeadG = LeadDataG[2,i];  // determine the coefficient of G
      i=ncols(LeadDataG)+1;
    }
  }

  G=LeadDataFm[1]*G-LeadG*Fm;  // leading terms in y should cancel...

  coef_Mat = coef(G,xy);
  Jnew=coef_Mat[2,1..ncols(coef_Mat)];

  // simplification of Jnew

  if (defined(artin_bd)) // the artin_bd-th power of the maxideal of
                         // deformation parameters can be cutted off
  {
    Jnew=jet(Jnew,artin_bd-1);
  }
  Jnew=interred(Jnew);
  if (defined(artin_bd)) { Jnew=jet(Jnew,artin_bd-1); }
  J=J,Jnew;

  if (typ==1) // isEquising
  {
    if(ideal(nselect(J,1,no_b))<>0)
    {
      setring old_ring;
      option(set,ov);
      return(0);
    }
  }

  while (fertig<>soll and blowup<nrows(M[3]))
  {
    upper_bound_old=upper_bound;
    dbprint(i_print,"// Blowup Step "+string(blowup)+" completed");
    blowup=blowup+1;

    for (branch=1;branch<=number_of_branches;branch=branch+1)
    {
      Jnew=0;

      // First we check if the branch still has to be considered:
      if (branch==upper_bound_old[branch] and fertig[branch]<>1)
      {
        if (M[3][blowup-1,branch]==1 and
               ((B[blowup,branch]<>x and B[blowup,branch]<>y)
            or (blowup==nrows(M[3])) ))
        {
          fertig[branch]=1;
          dbprint(i_print,"// 1 branch finished");
        }
      }

      if (branch<=upper_bound_old[branch] and fertig[branch]<>1)
      {
        for (i=branch;i>=1;i--)
        {
          if (M[1][blowup-1,i]<>0)
          {
            m=M[1][blowup-1,i]; // multiplicity before blowup
            i=0;
          }
        }

        // we blow up the branch and take the strict transform:
        attrib(J,"isSB",1);
        bl_Map(branch)=reduce(bl_Map(branch),J);
        attrib(J,"isSB",0);

        phi=myRing,bl_Map(branch);
        F(branch)=phi(F(branch))/x^m;

        // simplify F
        attrib(Jnew,"isSB",1);

        F(branch)=reduce(F(branch),Jnew);
        attrib(Jnew,"isSB",0);

        m=M[1][blowup,branch]; // multiplicity after blowup
        Fm=m_Jet(F(branch),m); // homogeneous part of lowest degree


        // we check for Fm=F[k]*...*F[k+s] where
        //
        //    F[j]=(y-b'(j)*x)^m(j), respectively F[j]=(-b'(j)*y+x)^m(j)
        //
        // according to the entries m(j)= M[3][blowup,j] and
        //                          b'(j) mod m_A = B[blowup,j]
        // computed from the HNE of the special fibre of the family:
        G=1;
        counter=branch;
        k=upper_bound[branch];

        F_save=F(branch);

        while(counter<=k)
        {
          F(counter)=m_Jet(F_save,maxDeg[blowup,counter]);

          if (B[blowup,counter]<>x and B[blowup,counter]<>y)
          {
            G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]);
            bl_Map(counter)=maxideal(1);
            bl_Map(counter)[nvars(basering)]=
                                        xy+(b(auxVar)+B[blowup,counter])*x;
            bNodes[counter]=b(auxVar);
            auxVar=auxVar+1;
          }
          else
          {
            if (B[blowup,counter]==x)
            {
              G=G*x^(M[3][blowup,counter]);  // branch has tangent x !!
              F(counter)=swapXY(F(counter)); // will turn x to y for blow up
              bl_Map(counter)=maxideal(1);
              bl_Map(counter)[nvars(basering)]=xy;
            }
            else
            {
              G=G*y^(M[3][blowup,counter]); // tangent has to be y
              bl_Map(counter)=maxideal(1);
              bl_Map(counter)[nvars(basering)]=xy;
            }
          }
          upper_bound[counter]=counter+M[2][blowup+1,counter]-1;
          counter=counter+M[2][blowup+1,counter];
        }
        G=determine_coef(Fm)[1]*G-Fm;  // leading terms in y should cancel
        coef_Mat = coef(G,xy);
        Jnew=coef_Mat[2,1..ncols(coef_Mat)];
        if (defined(artin_bd)) // the artin_bd-th power of the maxideal of
                              // deformation parameters can be cutted off
        {
          Jnew=jet(Jnew,artin_bd-1);
        }

        // simplification of J
        Jnew=interred(Jnew);

        J=J,Jnew;
        if (typ==1) // isEquising
        {
          if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); }
          if(ideal(nselect(J,1,no_b))<>0)
          {
            setring old_ring;
            option(set,ov);
            return(0);
          }
        }
      }
    }
    if (number_of_branches>=2)
    {
      J=interred(J);
      if (typ==1) // isEquising
      {
        if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); }
        if(ideal(nselect(J,1,no_b))<>0)
        {
          setring old_ring;
          option(set,ov);
          return(0);
        }
      }
    }
  }
 
  // Computation for all equimultiple sections being trivial (I^s(f))
  ideal Jtriv=J;
  for (i=1;i<=no_b; i++)
  {
    if (reduce(b(i),std(bNodes))!=0){
      Jtriv=subst(Jtriv,b(i),0);
    }
  }
  Jtriv=std(Jtriv);



  dbprint(i_print,"// ");
  dbprint(i_print,"// Elimination starts:");
  dbprint(i_print,"// -------------------");

  poly gg;
  int b_left=no_b;

  for (i=1;i<=no_b; i++)
  {
    attrib(J,"isSB",1);
    gg=reduce(b(i),J);
    if (gg==0)
    {
      b_left = b_left-1;  // another b(i) has to be 0
    }
    J = subst(J, b(i), gg);
    attrib(J,"isSB",0);
  }
  J=simplify(J,10);
  if (typ==1) // isEquising
  {
    if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); }
    if(ideal(nselect(J,1,no_b))<>0)
    {
      setring old_ring;
      option(set,ov);
      return(0);
    }
  }

  //new CL 11/06:  check in which equations b(k) appears and remove those b(k) 
  //               which appear in exactly one of the equations (by removing this
  //               equation)
  dbprint(i_print,"// ");
  dbprint(i_print,"// Remove superfluous equations:");
  dbprint(i_print,"// -----------------------------");
  int Z,App_in;
  ideal J_Tmp;
  int ncJ=ncols(J);

  intmat Mdet[ncJ][1];
  for (Z=1;Z<=ncJ;Z++){ Mdet[Z,1]=Z; }

  for (i=1;i<=no_b; i++)
  {
    ideal b_appears_in(i);            // Eintraege sind spaeter 1 oder 0 
    intmat b_app_in(i)[1][ncJ];       // Eintraege sind spaeter 1 oder 0 
    b_appears_in(i)[ncJ]=0;  
    J_Tmp = J-subst(J,b(i),0);
    for (Z=1; Z<=ncJ; Z++) { 
      if (J_Tmp[Z]<>0) {      // b(i) appear in J_Tmp[Z]
        b_appears_in(i)[Z]=1;
        b_app_in(i)[1,Z]=1;
      }
    }
    if (size(b_appears_in(i))==1) { //b(i) appears only in one J_Tmp[Z]
      App_in = (b_app_in(i)*Mdet)[1,1];  // determines Z
      J[App_in]=0;
      b_appears_in(i)[App_in]=0;
      b_app_in(i)[1,App_in]=0;
    }
  }

  for (i=1;i<=no_b; i++)
  {
    if (size(b_appears_in(i))==1) { //b(i) appears only in one J_Tmp[Z]
      App_in = (b_app_in(i)*Mdet)[1,1];  // determines Z
      J[App_in]=0;
      b_appears_in(i)[App_in]=0;
      b_app_in(i)[1,Z]=1;
      i=0;
    }
  }

  Jtriv = nselect(Jtriv,1,no_b);
  ideal J_no_b = nselect(J,1,no_b);
  if (size(J) > size(J_no_b))
  {
    dbprint(i_print,"// std computation started");
    // some b(i) didn't appear in linear conditions and have to be eliminated
    if (defined(artin_bd))
    {
      // first we make the ring smaller (removing variables, which are
      // forced to 0 by J
      list LL=make_ring_small(J);
      ideal Shortmap=LL[2];
      minPolyStr = "";
      if (minpoly !=0)
      {
        minPolyStr = "minpoly = "+string(minpoly);
      }
      ordStr = "dp(" + string(b_left) + "),dp";
      ideal qId = ideal(basering);

      helpStr = "ring Shortring = ("
                + charstr(basering) + "),("+ LL[1] +") , ("+ ordStr  +");";
      execute(helpStr);
      execute(minPolyStr);
      // ring has changed to "Shortring"

      ideal MM=maxideal(artin_bd);
      MM=subst(MM,x,0);
      MM=subst(MM,y,0);
      MM=simplify(MM,2);
      dbprint(i_print-1,"// maxideal("+string(artin_bd)+") has "
                         +string(size(MM))+" elements");
      dbprint(i_print-1,"//");

      // we change to the qring mod m^artin_bd
      // first, we have to check if we were in a qring when starting
      ideal qId = imap(myRing, qId);
      if (qId == 0)
      {
         attrib(MM,"isSB",1);
         qring QQ=MM;
      }
      else
      {
         qId=qId,MM;
         qring QQ = std(qId);
      }

      ideal Shortmap=imap(myRing,Shortmap);
      map phiphi=myRing,Shortmap;

      ideal J=phiphi(J);
      option(redSB);
      J=std(J);
      J=nselect(J,1,no_b);

      setring myRing;
      // back to "myRing"

      J=nselect(J,1,no_b);
      Jnew=imap(QQ,J);

      J=J,Jnew;
      J=interred(J);
      if (defined(artin_bd)){ J=jet(J,artin_bd-1); }
    }
    else
    {
      J=std(J);
      J=nselect(J,1,no_b);
      if (defined(artin_bd)){ J=jet(J,artin_bd-1); }
    }
  }

  dbprint(i_print,"// finished");
  dbprint(i_print,"// ");

  minPolyStr = "";option(set,ov);
  if (minpoly !=0)
  {
   minPolyStr = "minpoly = "+string(minpoly);
  }

  kill HNEring;

  if (typ==1) // isEquising
  {
    if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); }
    if(J<>0)
    {
      setring old_ring;
      option(set,ov);
      return(0);
    }
    else
    {
      setring old_ring;
      option(set,ov);
      return(1);
    }
  }

  setring old_ring;
  // we are back in the original ring

  if (npars(myRing)<>0)
  {
    ideal qIdeal = ideal(basering);
    helpStr = "ring ESSring = ("
                 + string(char(basering))+ "," + parstr(myRing) +
                 ") , ("+ varstr(basering)+") , ("+ ordstr(basering) +");";
    execute(helpStr);
    execute(minPolyStr);
    // basering has changed to ESSring

    ideal qIdeal = fetch(old_ring, qIdeal);
    if(qIdeal != 0)
    {
      def r_base = basering;
      kill ESSring;
      qring ESSring = std(qIdeal);
    }
    kill qIdeal;

    ideal SSS;
    for (int ii=1;ii<=nvars(basering);ii++)
    {
      SSS[ii+no_b]=var(ii);
    }
    map phi=myRing,SSS;   // b(i) variables are mapped to zero

    ideal ES=phi(J);
    ideal ES_all_triv=phi(Jtriv);
    kill phi;

    if (defined(p_F)<=0)
    {
      poly p_F=fetch(old_ring,p_F);
      export(p_F);
    }
    export(ES);
    export(ES_all_triv);
    setring old_ring;
    dbprint(i_print+2,"
// 'esStratum' created a list M of a ring and an integer.
// To access the ideal defining the equisingularity stratum, type:
        def ESSring = M[1]; setring ESSring;  ES; ");

    option(set,ov);
    return(list(ESSring,0));
  }
  else
  {
    // no new ring definition necessary
    ideal SSS;
    for (int ii=1;ii<=nvars(basering);ii++)
    {
      SSS[ii+no_b]=var(ii);
    }
    map phi=myRing,SSS;  // b(i) variables are mapped to zero

    ideal ES=phi(J);
    ideal ES_all_triv=phi(Jtriv);
    kill phi;

    setring old_ring;
    dbprint(i_print,"// output of 'esStratum' is a list consisting of:
//    _[1][1] = ideal defining the equisingularity stratum
//    _[1][2] = ideal defining the part of the equisingularity stratum
//              where all equimultiple sections are trivial
//    _[2] = 0");

    option(set,ov);
    return(list(list(ES,ES_all_triv),0));
  }

}

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

proc tau_es (poly f,list #)
"USAGE:   tau_es(f); f poly
ASSUME:  f is a reduced bivariate polynomial, the basering has precisely
         two variables, is local and no qring.
RETURN:  int, the codimension of the mu-const stratum in the semi-universal
         deformation base.
NOTE:    printlevel>=1 displays additional information.
         When called with any additional parameter, the computation of the
         Milnor number is avoided (no check for NND).
SEE ALSO: esIdeal, tjurina, invariants
EXAMPLE: example tau_es; shows an example.
"
{
  int i,j,k,s;
  int slope_x, slope_y, upper;
  int i_print = printlevel - voice + 3;
  string MinPolyStr;

  // some checks first
  if ( nvars(basering)<>2 )
  {
    print("// basering has not the correct number (two) of variables !");
    print("// computation stopped");
    return(0);
  }
  if ( mult(std(1+var(1)+var(2))) <> 0)
  {
    print("// basering is not local !");
    print("// computation stopped");
    return(0);
  }

  if (mult(std(f))<=1)
  {
    // f is rigid
    return(0);
  }

  if ( deg(squarefree(f))!=deg(f) )
  {
    print("// input polynomial was not reduced");
    print("// try    squarefree(f);   first");
    return(0);
  }

  def old_ring=basering;
  execute("ring @myRing=("+charstr(basering)+"),("+varstr(basering)+"),ds;");
  poly f=imap(old_ring,f);

  ideal Jacobi_Id = jacob(f);

  // check for A_k singularity
  // ----------------------------------------
  if (mult(std(f))==2)
  {
    dbprint(i_print-1,"// ");
    dbprint(i_print-1,"// polynomial defined A_k singularity");
    dbprint(i_print-1,"// ");
    return( vdim(std(Jacobi_Id)) );
  }

  // check for D_k singularity
  // ----------------------------------------
  if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3)
  {
    dbprint(i_print,"// ");
    dbprint(i_print,"// polynomial defined D_k singularity");
    dbprint(i_print,"// ");
    ideal ES_Id = f, jacob(f);
    return( vdim(std(Jacobi_Id)));
  }


  if (size(#)==0)
  {
    // check if Newton polygon non-degenerate
    // ----------------------------------------
    Jacobi_Id=std(Jacobi_Id);
    int mu = vdim(Jacobi_Id);
    poly f_tilde=f+var(1)^mu+var(2)^mu;  //to obtain convenient Newton-polygon

    list NP=newtonpoly(f_tilde);
    dbprint(i_print-1,"// Newton polygon:");
    dbprint(i_print-1,NP);
    dbprint(i_print-1,"");

    if(is_NND(f,mu,NP))          // f is Newton non-degenerate
    {
      upper=NP[1][2];
      ideal ES_Id= x^k*y^upper;
      dbprint(i_print-1,"polynomial is Newton non-degenerate");
      dbprint(i_print-1,"");
      k=0;
      for (i=1;i<=size(NP)-1;i++)
      {
        slope_x=NP[i+1][1]-NP[i][1];
        slope_y=NP[i][2]-NP[i+1][2];
        for (k=NP[i][1]+1; k<=NP[i+1][1]; k++)
        {
          while ( slope_x*upper + slope_y*k >=
                  slope_x*NP[i][2] + slope_y*NP[i][1])
          {
            upper=upper-1;
          }
          upper=upper+1;
          ES_Id=ES_Id, x^k*y^upper;
        }
      }
      ES_Id=std(ES_Id);
      dbprint(i_print-2,"ideal of monomials above Newton bd. is generated by:");
      dbprint(i_print-2,ES_Id);
      ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f);
      ES_Id = ES_Id, Jacobi_Id;
      ES_Id = std(ES_Id);
      dbprint(i_print-1,"// ");
      dbprint(i_print-1,"// Equisingularity ideal is computed!");
      dbprint(i_print-1,"");
      return(vdim(ES_Id));
    }
    else
    {
      dbprint(i_print-1,"polynomial is Newton degenerate !");
      dbprint(i_print-1,"");
    }
  }

  // for Newton degenerate polynomials, we compute the HN expansion, and
  // count the number of free points .....

  dbprint(i_print-1,"// ");
  dbprint(i_print-1,"// Compute HN expansion");
  dbprint(i_print-1,"// ---------------------");
  i=printlevel;
  printlevel=printlevel-5;
  if (2*size(coeffs(f,x))<size(coeffs(f,y)))
  {
    f=swapXY(f);
  }
  list LLL=hnexpansion(f);
  if (size(LLL)==0) { // empty list returned by hnexpansion
    setring old_ring;
    ERROR("Unable to compute HN expansion !");
  }
  else
  {
    if (typeof(LLL[1])=="ring") {
      def HNering = LLL[1];
      setring HNering;
      def @L=hne;
    }
    else {
      def @L=LLL;
    }
  }
  def HNEring=basering;

  printlevel=i;
  dbprint(i_print-1,"// finished");
  dbprint(i_print-1,"// ");

  list M=multsequence(@L);
  M=control_Matrix(M);     // this returns the 4 control matrices
  intmat Mult=M[1];

  list L1=inf_Tangents(@L,nrows(M[1]));
  matrix B=L1[1];

  // determine sum_i m_i(m_i+1)/2 (over inf. near points)
  int conditions=0;
  for (i=1;i<=nrows(Mult);i++)
  {
    for (j=1;j<=ncols(Mult);j++)
    {
      conditions=conditions+(Mult[i,j]*(Mult[i,j]+1)/2);
    }
  }
  int freePts=no_freePoints(M[1],B);
  int taues=conditions-freePts-2;

  setring old_ring;
  return(taues);
}
example
{
   "EXAMPLE:"; echo=2;
   ring r=32003,(x,y),ds;
   poly f=(x4-y4)^2-x10;
   tau_es(f);
}


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

proc esIdeal (poly f,list #)
"USAGE:   esIdeal(f[,any]]); f poly
ASSUME:  f is a reduced bivariate polynomial, the basering has precisely
         two variables, is local and no qring, and the characteristic of
         the ground field does not divide mult(f).
RETURN:  if called with only one parameter: list of two ideals,
@format
          _[1]:  equisingularity ideal of f (in sense of Wahl),
          _[2]:  ideal of equisingularity with fixed position of the
                 singularity;
@end format
         if called with more than one parameter: list of three ideals,
@format
          _[1]:  equisingularity ideal of f (in sense of Wahl)
          _[2]:  ideal of equisingularity with fixed position of the
                 singularity;
          _[3]:  ideal of all g such that the deformation defined by f+eg
                 (e^2=0) is isomorphic to an equisingular deformation
                 of V(f) with all equimultiple sections being trivial.
@end format
NOTE:    if some of the above condition is not satisfied then return
         value is list(0,0).
SEE ALSO: tau_es, esStratum
KEYWORDS: equisingularity ideal
EXAMPLE: example esIdeal; shows examples.
"
{

  int typ;
  if (size(#)>0) { typ=1; }   // I^s is also computed
  int i,k,s;
  int slope_x, slope_y, upper;
  int i_print = printlevel - voice + 3;
  string MinPolyStr;

  // some checks first
  if ( nvars(basering)<>2 )
  {
    print("// basering has not the correct number (two) of variables !");
    print("// computation stopped");
    return(list(0,0));
  }
  if ( mult(std(1+var(1)+var(2))) <> 0)
  {
    print("// basering is not local !");
    print("// computation stopped");
    return(list(0,0));
  }

  if (mult(std(f))<=1)
  {
    // f is rigid
    if (typ==0)
    {
      return(list(ideal(1),ideal(1)));
    }
    else
    {
      return(list(ideal(1),ideal(1),ideal(1)));
    }
  }

  if ( deg(squarefree(f))!=deg(f) )
  {
    print("// input polynomial was not squarefree");
    print("// try    squarefree(f);   first");
    return(list(0,0));
  }

  if (char(basering)<>0)
  {
    if (mult(std(f)) mod char(basering)==0)
    {
      print("// characteristic of ground field divides "
            + "multiplicity of polynomial !");
      print("// computation stopped");
      return(list(0,0));
    }
  }

  // check for A_k singularity
  // ----------------------------------------
  if (mult(std(f))==2)
  {
    dbprint(i_print,"// ");
    dbprint(i_print,"// polynomial defined A_k singularity");
    dbprint(i_print,"// ");
    ideal ES_Id = f, jacob(f);
    ES_Id = interred(ES_Id);
    ideal ESfix_Id = f, maxideal(1)*jacob(f);
    ESfix_Id= interred(ESfix_Id);
    if (typ==0) // only for computation of I^es and I^es_fix
    {
      return( list(ES_Id,ESfix_Id) );
    }
    else
    {
      return( list(ES_Id,ESfix_Id,ES_Id) );
    }
  }

  // check for D_k singularity
  // ----------------------------------------
  if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3)
  {
    dbprint(i_print,"// ");
    dbprint(i_print,"// polynomial defined D_k singularity");
    dbprint(i_print,"// ");
    ideal ES_Id = f, jacob(f);
    ES_Id = interred(ES_Id);
    ideal ESfix_Id = f, maxideal(1)*jacob(f);
    ESfix_Id= interred(ESfix_Id);
    if (typ==0) // only for computation of I^es and I^es_fix
    {
      return( list(ES_Id,ESfix_Id) );
    }
    else
    {
      return( list(ES_Id,ESfix_Id,ES_Id) );
    }
  }

  // check if Newton polygon non-degenerate
  // ----------------------------------------
  int mu = milnor(f);
  poly f_tilde=f+var(1)^mu+var(2)^mu;  //to obtain a convenient Newton-polygon

  list NP=newtonpoly(f_tilde);
  dbprint(i_print-1,"// Newton polygon:");
  dbprint(i_print-1,NP);
  dbprint(i_print-1,"");

  if(is_NND(f,mu,NP))          // f is Newton non-degenerate
  {
    upper=NP[1][2];
    ideal ES_Id= x^k*y^upper;
    dbprint(i_print,"polynomial is Newton non-degenerate");
    dbprint(i_print,"");
    k=0;
    for (i=1;i<=size(NP)-1;i++)
    {
      slope_x=NP[i+1][1]-NP[i][1];
      slope_y=NP[i][2]-NP[i+1][2];
      for (k=NP[i][1]+1; k<=NP[i+1][1]; k++)
      {
        while ( slope_x*upper + slope_y*k >=
                slope_x*NP[i][2] + slope_y*NP[i][1])
        {
          upper=upper-1;
        }
        upper=upper+1;
        ES_Id=ES_Id, x^k*y^upper;
      }
    }
    ES_Id=std(ES_Id);
    dbprint(i_print-1,"ideal of monomials above Newton bd. is generated by:");
    dbprint(i_print-1,ES_Id);
    ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f);
    ES_Id = ES_Id, f, jacob(f);
    dbprint(i_print,"// ");
    dbprint(i_print,"// equisingularity ideal is computed!");
    if (typ==0)
    {
      return(list(ES_Id,ESfix_Id));
    }
    else
    {
      return(list(ES_Id,ESfix_Id,ES_Id));
    }
  }
  else
  {
    dbprint(i_print,"polynomial is Newton degenerate !");
    dbprint(i_print,"");
  }

  def old_ring=basering;

  dbprint(i_print,"// ");
  dbprint(i_print,"// versal deformation with triv. section");
  dbprint(i_print,"// =====================================");
  dbprint(i_print,"// ");

  ideal JJ=maxideal(1)*jacob(f);
  ideal kbase_versal=kbase(std(JJ));
  s=size(kbase_versal);
  string ring_versal="ring @Px = ("+charstr(basering)+"),(t(1.."+string(s)+"),"
                        +varstr(basering)+"),(ds("+string(s)+"),"
                        +ordstr(basering)+");";
  MinPolyStr = string(minpoly);

  execute(ring_versal);
  if (MinPolyStr<>"0")
  {
    MinPolyStr = "minpoly="+MinPolyStr;
    execute(MinPolyStr);
  }
  // basering has changed to @Px

  poly F=imap(old_ring,f);
  ideal kbase_versal=imap(old_ring,kbase_versal);
  for (i=1; i<=s; i++)
  {
    F=F+var(i)*kbase_versal[i];
  }
  dbprint(i_print-1,F);
  dbprint(i_print-1,"");


  ideal ES_Id,ES_Id_all_triv;
  poly Ftriv=F;

  dbprint(i_print,"// ");
  dbprint(i_print,"// Compute equisingularity Stratum over Spec(C[t]/t^2)");
  dbprint(i_print,"// ===================================================");
  dbprint(i_print,"// ");
  list M=esStratum(F,2);
  dbprint(i_print,"// finished");
  dbprint(i_print,"// ");

  if (M[2]==1) // error occured during esStratum computation
  {
    print("Some error has occured during the computation");
    return(list(0,0));
  }

  if ( typeof(M[1])=="list" )
  {
    int defpars = nvars(basering)-2;
    poly Fred,Ftrivred;
    poly g;
    F=reduce(F,std(M[1][1]));
    Ftriv=reduce(Ftriv,std(M[1][2]));

    for (i=1; i<=defpars; i++)
    {
      Fred=reduce(F,std(var(i)));
      Ftrivred=reduce(Ftriv,std(var(i)));

      g=subst(F-Fred,var(i),1);
      ES_Id=ES_Id, g;
      F=Fred;

      g=subst(Ftriv-Ftrivred,var(i),1);
      ES_Id_all_triv=ES_Id_all_triv, g;
      Ftriv=Ftrivred;
    }

    setring old_ring;
    // back to original ring

    ideal ES_Id = imap(@Px,ES_Id);
    ES_Id = interred(ES_Id);

    ideal ES_Id_all_triv = imap(@Px,ES_Id_all_triv);
    ES_Id_all_triv = interred(ES_Id_all_triv);

    ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f);
    ES_Id = ES_Id, f, jacob(f);
    ES_Id_all_triv = ES_Id_all_triv, f, jacob(f);

    if (typ==0)
    {
      return(list(ES_Id,ESfix_Id));
    }
    else
    {
      return(list(ES_Id,ESfix_Id,ES_Id_all_triv));
    }
  }
  else
  {
    def AuxRing=M[1];

    dbprint(i_print,"// ");
    dbprint(i_print,"// change ring to ESSring");

    setring AuxRing;  // contains p_F, ES

    int defpars = nvars(basering)-2;
    poly Fred,Fredtriv;
    poly g;
    ideal ES_Id,ES_Id_all_triv;

    poly p_Ftriv=p_F;

    p_F=reduce(p_F,std(ES));
    p_Ftriv=reduce(p_Ftriv,std(ES_all_triv));
    for (i=1; i<=defpars; i++)
    {
      Fred=reduce(p_F,std(var(i)));
      Fredtriv=reduce(p_Ftriv,std(var(i)));

      g=subst(p_F-Fred,var(i),1);
      ES_Id=ES_Id, g;
      p_F=Fred;

      g=subst(p_Ftriv-Fredtriv,var(i),1);
      ES_Id_all_triv=ES_Id_all_triv, g;
      p_Ftriv=Fredtriv;

    }

    dbprint(i_print,"// ");
    dbprint(i_print,"// back to the original ring");

    setring old_ring;
    // back to original ring

    ideal ES_Id = imap(AuxRing,ES_Id);
    ES_Id = interred(ES_Id);

    ideal ES_Id_all_triv = imap(AuxRing,ES_Id_all_triv);
    ES_Id_all_triv = interred(ES_Id_all_triv);

    kill @Px;
    kill AuxRing;

    ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f);
    ES_Id = ES_Id, f, jacob(f);
    ES_Id_all_triv = ES_Id_all_triv, f, jacob(f);
    dbprint(i_print,"// ");
    dbprint(i_print,"// equisingularity ideal is computed!");
    if (typ==0)
    {
      return(list(ES_Id,ESfix_Id));
    }
    else
    {
      return(list(ES_Id,ESfix_Id,ES_Id_all_triv));
    }
  }
}
example
{
  "EXAMPLE:"; echo=2;
  ring r=0,(x,y),ds;
  poly f=x7+y7+(x-y)^2*x2y2;
  list K=esIdeal(f);
  option(redSB);
  // Wahl's equisingularity ideal:
  std(K[1]);

  ring rr=0,(x,y),ds;
  poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31;
  list K=esIdeal(f);
  vdim(std(K[1]));
  // the latter should be equal to:
  tau_es(f);
}

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

proc esStratum (poly p_F, list #)
"USAGE:   esStratum(F[,m,L]); F poly, m int, L list
ASSUME:  F defines a deformation of a reduced bivariate polynomial f
         and the characteristic of the basering does not divide mult(f). @*
         If nv is the number of variables of the basering, then the first
         nv-2 variables are the deformation parameters. @*
         If the basering is a qring, ideal(basering) must only depend
         on the deformation parameters.
COMPUTE: equations for the stratum of equisingular deformations with
         fixed (trivial) section.
RETURN:  list l: either consisting of a list and an integer, where
@format
  l[1][1]=ideal defining the equisingularity stratum
  l[1][2]=ideal defining the part of the equisingularity stratum where all
          equimultiple sections through the non-nodes of the reduced total
          transform are trivial sections
  l[2]=1 if some error has occured,  l[2]=0 otherwise;
@end format
or consisting of a ring and an integer, where
@format
  l[1]=ESSring is a ring extension of basering containing the ideal ES
        (describing the ES-stratum), the ideal ES_all_triv (describing the
        part with trival equimultiple sections) and the poly p_F=F,
  l[2]=1 if some error has occured,  l[2]=0 otherwise.
@end format
NOTE:    L is supposed to be the output of hnexpansion (with the given ordering
         of the variables appearing in f). @*
         If m is given, the ES Stratum over A/maxideal(m) is computed. @*
         This procedure uses @code{execute} or calls a procedure using
         @code{execute}.
         printlevel>=2 displays additional information.
SEE ALSO: esIdeal, isEquising
KEYWORDS: equisingularity stratum
EXAMPLE: example esStratum; shows examples.
"
{
  list l=esComputation (0,p_F,#);
  return(l);
}
example
{
   "EXAMPLE:"; echo=2;
   int p=printlevel;
   printlevel=1;
   ring r = 0,(a,b,c,d,e,f,g,x,y),ds;
   poly F = (x2+2xy+y2+x5)+ax+by+cx2+dxy+ey2+fx3+gx4;
   list M = esStratum(F);
   M[1][1];

   printlevel=3;     // displays additional information
   esStratum(F,2)  ; // ES-stratum over Q[a,b,c,d,e,f,g] / <a,b,c,d,e,f,g>^2

   ideal I = f-fa,e+b;
   qring q = std(I);
   poly F = imap(r,F);
   esStratum(F);
   printlevel=p;
}

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

proc isEquising (poly p_F, list #)
"USAGE:   isEquising(F[,m,L]); F poly, m int, L list
ASSUME:  F defines a deformation of a reduced bivariate polynomial f
         and the characteristic of the basering does not divide mult(f). @*
         If nv is the number of variables of the basering, then the first
         nv-2 variables are the deformation parameters. @*
         If the basering is a qring, ideal(basering) must only depend
         on the deformation parameters.
COMPUTE: tests if the given family is equisingular along the trivial
         section.
RETURN:  int: 1 if the family is equisingular, 0 otherwise.
NOTE:    L is supposed to be the output of hnexpansion (with the given ordering
         of the variables appearing in f). @*
         If m is given, the family is considered over A/maxideal(m). @*
         This procedure uses @code{execute} or calls a procedure using
         @code{execute}.
         printlevel>=2 displays additional information.
EXAMPLE: example isEquising; shows examples.
"
{
  int check=esComputation (1,p_F,#);
  return(check);
}
example
{
   "EXAMPLE:"; echo=2;
   ring r = 0,(a,b,x,y),ds;
   poly F = (x2+2xy+y2+x5)+ay3+bx5;
   isEquising(F);
   ideal I = ideal(a);
   qring q = std(I);
   poly F = imap(r,F);
   isEquising(F);

   ring rr=0,(A,B,C,x,y),ls;
   poly f=x7+y7+(x-y)^2*x2y2;
   poly F=f+A*y*diff(f,x)+B*x*diff(f,x);
   isEquising(F);
   isEquising(F,2);    // computation over  Q[a,b] / <a,b>^2
}



/*  Examples:

LIB "equising.lib";
   ring r = 0,(x,y),ds;
   poly p1 = y^2+x^3;
   poly p2 = p1^2+x5y;
   poly p3 = p2^2+x^10*p1;
   poly p=p3^2+x^20*p2;
   p;
   list L=versal(p);
   def Px=L[1];
   setring Px;
   poly F=Fs[1,1];
   int t=timer;
   list M=esStratum(F);
   timer-t;  //-> 3

LIB "equising.lib";
option(prot);
printlevel=2;
ring r=0,(x,y),ds;
poly f=(x-yx+y2)^2-(y+x)^31;
list L=versal(f);
def Px=L[1];
setring Px;
poly F=Fs[1,1];
int t=timer;
list M=esStratum(F);
timer-t;  //-> 233


LIB "equising.lib";
printlevel=2;
option(prot);
timer=1;
ring r=0,(x,y),ls;
poly f=(x4-y4)^2-x10;
list L=versal(f);
def Px=L[1];
setring Px;
poly F=Fs[1,1];
int t=timer;
list M=esStratum(F,3);
timer-t;  //-> 8

LIB "equising.lib";
printlevel=2;
timer=1;
ring rr=0,(x,y),ls;
poly f=x7+y7+(x-y)^2*x2y2;
list K=esIdeal(f);
// tau_es
vdim(std(K[1])); //-> 22
// tau_es_fix
vdim(std(K[2])); //-> 24


LIB "equising.lib";
printlevel=2;
timer=1;
ring rr=0,(x,y),ls;
poly f=x7+y7+(x-y)^2*x2y2+x2y4; // Newton non-deg.
list K=esIdeal(f);
// tau_es
vdim(std(K[1])); //-> 21
// tau_es_fix
vdim(std(K[2])); //-> 23

LIB "equising.lib";
ring r=0,(w,v),ds;
poly f=w2-v199;
list L=hnexpansion(f);
list LL=versal(f);
def Px=LL[1];
setring Px;
list L=imap(r,L);
poly F=Fs[1,1];
list M=esStratum(F,2,L);

LIB "equising.lib";
printlevel=2;
timer=1;
ring rr=0,(A,B,C,x,y),ls;
poly f=x7+y7+(x-y)^2*x2y2;
poly F=f+A*y*diff(f,x)+B*x*diff(f,x)+C*diff(f,y);
list M=esStratum(F,6);
std(M[1][1]);  // standard basis of equisingularity ideal

LIB "equising.lib";
printlevel=2;
timer=1;
ring rr=0,(x,y),ls;
poly f=x20+y7+(x-y)^2*x2y2+x2y4; // Newton non-degenerate
list K=esIdeal(f);
K;

ring rr=0,(x,y),ls;
poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13;
list K=esIdeal(f);
list L=versal(f);
def Px=L[1];
setring Px;
poly F=Fs[1,1];
list M=esStratum(F,2);

LIB "equising.lib";
ring R=0,(A,B,C,D,x,y),ds;
poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13;
poly F=f+Ax9+Bx7y2+Cx9y+Dx8y2;
list M=esStratum(F,2);


LIB "equising.lib";
printlevel=2;
ring rr=0,(x,y),ls;
poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13;
list K=esIdeal(f);
vdim(std(K[1]));  //-> 51
tau_es(f);        //-> 51

printlevel=3;
f=f*(y-x2)*(y2-x3)*(x-y5);
int t=timer;
list L=esIdeal(f);
vdim(std(L[1]));  //-> 99
timer-t;   //-> 42
t=timer;
tau_es(f);        //-> 99
timer-t;   //-> 23


LIB "equising.lib";
printlevel=3;
ring rr=0,(x,y),ds;
poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31;
list K=esIdeal(f);
vdim(std(K[1]));  //-> 68
tau_es(f);        //-> 68

list L=versal(f);
def Px=L[1];
setring Px;
poly F=Fs[1,1];
list M=esStratum(F);
timer-t;          //-> 0



*/