source: git/Singular/LIB/qhmoduli.lib @ 70597c

///////////////////////////////////////////////////////////////////////////////
version="Id: qhmoduli.lib,v 1.0 2000/12/12 12:32:15 Singular Exp $";
category="Singularities";
info="
LIBRARY:  qhmoduli.lib    Moduli Spaces of Semi-Quasihomogeneous Singularities
AUTHOR:   Thomas Bayer, email: bayert@in.tum.de

PROCEDURES:
 ArnoldAction(f, [G, w])  Induced action of G_f on T_.
 ModEqn(f)                Equations of the moduli space for principal part f
 QuotientEquations(G,A,I) Equations of Variety(I)/G w.r.t. action 'A'
 StabEqn(f)               Equations of the stabilizer of f.
 StabEqnId(I, w)          Equations of the stabilizer of the qhom. ideal I.
 StabOrder(f)             Order of the stabilizer of f.
 UpperMonomials(f, [w])   Upper basis of the Milnor algebra of f.

 Max(data)                maximal integer contained in 'data'
 Min(data)                minimal integer contained in  'data'
";

// NOTE: This library has been written in the frame of the diploma thesis
// 'Computing moduli spaces of semiquasihomogeneous singularities and an
//  implementation in Singular', Arbeitsgruppe Algebraische Geometrie,
// Fachbereich Mathematik, University Kaiserslautern,
// Advisor: Prof. Gert-Martin Greuel

LIB "rinvar.lib";

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

proc ModEqn(poly f, list #)
"USAGE:   ModEqn(f [, opt]); poly f; int opt;
PURPOSE: compute equations of the moduli space of semiquasihomogenos hypersurface         singularity with principal part f w.r.t. right equivalence
ASSUME:  f quasihomogeneous polynomial with an isolated singularity at 0
RETURN:  polynomial ring, possibly a simple extension of the groundfield of
         the basering, containing the ideal 'modid'
         - 'modid' is the ideal of the moduli space  if opt is even (> 0).
           otherwise it contains generators of the coordinate ring R of the
           moduli space (note : Spec(R) is the moduli space)
OPTIONS: 1 compute equations of the mod. space,
         2 use a primary decomposition
         4 compute E_f0, i.e., the image of G_f0
         To combine options, add their value, default: opt =7
EXAMPLE: example ModEqn; shows an example
"
{
  int sizeOfAction, i, dimT, nonLinearQ, milnorNr, dbPrt;
  int imageQ, opt;
  intvec wt;
  ideal B;
  list Gf, tIndex, sList;
  string ringSTR;

  dbPrt = printlevel-voice+2;
  if(size(#) > 0) { opt = #[1]; }
  else { opt = 7; }
  if(opt / 4 > 0) { imageQ = 1; opt = opt - 4;}
  else { imageQ = 0; }

  wt = weight(f);
  milnorNr = vdim(std(jacob(f)));
  if(milnorNr == -1) {
                ERROR("the polynomial " + string(f) + " has a nonisolated singularity at 0");
        }       // singularity not isolated

  // 1st step : compute a basis of T_

  B = UpperMonomials(f, wt);
  dimT = size(B);
  dbprint(dbPrt, "moduli equations of f = " + string(f) + ", f has Milnor number = " + string(milnorNr));
  dbprint(dbPrt, " upper basis = " + string(B));
  if(size(B) > 1) {

    // 2nd step : compute the stabilizer G_f of f

    dbprint(dbPrt, " compute equations of the stabilizer of f, called G_f");
    Gf = StabEqn(f);
    dbprint(dbPrt, " order of the stabilizer = " + string(StabOrder(Gf)));

    // 3rd step : compute the induced action of G_f on T_ by means of a theorem of Arnold

    dbprint(dbPrt, " compute the induced action");
    def RME1 = ArnoldAction(f, Gf, B);
    setring(RME1);
    export(RME1);
    dbprint(dbPrt, " G_f = " + string(stabid));
    dbprint(dbPrt, " action of G_f : " + string(actionid));

    // 4th step : linearize the action of G_f

    sizeOfAction = size(actionid);
    def RME2 = LinearizeAction(stabid, actionid, nvars(Gf[1]));
    setring RME2;
    export(RME2);
    kill(RME1);

    if(size(actionid) == sizeOfAction) { nonLinearQ = 0;}
    else  {
      nonLinearQ = 1;
      dbprint(dbPrt, " linearized action = " + string(actionid));
      dbprint(dbPrt, " embedding of T_ = " + string(embedid));
    }



    if(!imageQ) {        // do not compute the image of Gf
      // 5th step : set E_f = G_f,
      dbprint(dbPrt, " compute equations of the quotient T_/G_f");
      def RME3 = basering;
    }
    else {

      // 5th step : compute the ideal and the action of E_f

      dbprint(dbPrt, " compute E_f");
      def RME3 = ImageGroup(groupid, actionid);
      setring(RME3);
      ideal embedid = imap(RME2, embedid);
      dbprint(dbPrt, " E_f  = (" + string(groupid) + ")");
      dbprint(dbPrt, " action of E'f = " + string(actionid));
      dbprint(dbPrt, " compute equations of the quotient T_/E_f");
    }
    export(RME3);
    kill(RME2);

    // 6th step : compute the equations of the quotient T_/E_f

                ideal G = groupid; ideal variety = embedid;
                kill(groupid); kill(embedid);
    def RME4 = QuotientEquations(G, actionid, variety, opt);
    setring RME4;
    poly mPoly = minpoly;
    kill(RME3);
    export(RME4);

    // simplify the ideal and create a new ring with propably less variables

    if(opt == 1 || opt == 3) {      // equations computed ?
      sList = SimplifyIdeal(id, 0, "Y");
      ideal newid = sList[1];
      dbprint(dbPrt, " number of equations = " + string(size(sList[1])));
      dbprint(dbPrt, " number of variables = " + string(size(sList[3])));
      ringSTR = "ring RME5 = (" + charstr(basering) + "), (Y(1.." + string(size(sList[3])) + ")),dp;";
      execute(ringSTR);
      minpoly = number(imap(RME4, mPoly));
      ideal modid = imap(RME4, newid);
    }
    else {
      def RME5 = RME4;
      setring(RME5);
      ideal modid = imap(RME4, id);
    }
    export(modid);
    kill(RME4);
  }
  else {
                def RME5 = basering;
                ideal modid = maxideal(1);
                if(size(B) == 1) {                      // 1-dimensional
                        modid[size(modid)] = 0;
                        modid = simplify(modid,2);
                }
                export(modid);
        }
dbprint(dbPrt, "
// 'ModEqn' created a new ring.
// To see the ring, type (if the name of the ring is R):
     show(R);
// To access the ideal of the moduli space of semiquasihomogeneous singularities
// with principal part f, type
     def R = ModEqn(f); setring R;  modid;
// 'modid' is the ideal of the moduli space.
// if 'opt' = 0 or even, then 'modid' contains algebra generators of S s.t.
// spec(S) = moduli space of f.
");
  return(RME5);
}
example
{"EXAMPLE:";  echo = 2;
  ring B   = 0,(x,y), ls;
  poly f = -x4 + xy5;
  def R = ModEqn(f);
  setring R;
  modid;
}


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

proc QuotientEquations(ideal G, ideal Gaction, ideal embedding, list#)
"USAGE:   QuotientEquations(G,action,emb [, opt]); ideal G,action,emb;int opt
PURPOSE: compute the quotient of the variety given by the parameterization
         'emb'  by the linear action 'action' of the algebraic group G.
ASSUME:  'action' is linear, G must be finite if the Reynolds operator is
         needed (i.e., NullCone(G,action) returns some non-invariant polys)
RETURN:   polynomial ring over a simple extension of the groundfield of the
          basering, containing the ideals 'id' and 'embedid'.
          - 'id' contains the equations of the quotient, if opt = 1;
            if opt = 0, 2, 'id' contains generators of the coordinate ring R
            of the quotient (Spec(R) is the quotient)
          - 'embedid' = 0, if opt = 1;
            if opt = 0, 2, it is the ideal defining the equivariant embedding
OPTIONS: 1 compute equations of the quotient,
         2 use a primary decomposition when computing the Reynolds operator @*
         To combine options, add their value, default: opt =3.
EXAMPLE: example QuotientEquations; shows an example
"
{
  int i, opt, primaryDec, relationsQ, dbPrt;
  ideal Gf, variety;
  intvec wt;

  dbPrt = printlevel-voice+3;
  if(size(#) > 0) { opt = #[1]; }
  else { opt = 3; }

  if(opt / 2 > 0) { primaryDec = 1; opt = opt - 2; }
  else { primaryDec = 0; }
  if(opt > 0) { relationsQ = 1;}
  else { relationsQ = 0; }

  Gf = std(G);
  variety = EquationsOfEmbedding(embedding, nvars(basering) - size(Gaction));

  if(size(variety) == 0) {    // use Hilbert function !
    for(i = 1; i <= ncols(Gaction); i ++) { wt[i] = 1;};
  }
  def RQER = InvariantRing(Gf, Gaction, primaryDec);    // compute the nullcone of the linear action

  def RQEB = basering;
  setring(RQER);
  export(RQER);

  if(relationsQ > 0) {
    dbprint(dbPrt, " compute equations of the variety (" + string(size(imap(RQER, invars))) + " invariants) ");
    if(!defined(variety)) { ideal variety = imap(RQEB, variety); }
    if(wt[1] > 0) {
      def RQES = ImageVariety(variety, imap(RQER, invars), wt);
    }
    else {
      def RQES = ImageVariety(variety, imap(RQER, invars));  // forget imap
    }
    setring(RQES);
    ideal id = imageid;
    ideal embedid = 0;
  }
  else {
    def RQES = basering;
    ideal id =  imap(RQER, invars);
    ideal embedid = imap(RQEB, variety);
  }
  kill(RQER);
  export(id);
  export(embedid);
  return(RQES);
}

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

proc UpperMonomials(poly f, list #)
"USAGE:   UpperMonomials(poly f, [intvec w])
PURPOSE: compute the upper monomials of the milnor algebra of f.
ASSUME:  f is quasihomogeneous (w.r.t. w)
RETURN:  ideal
EXAMPLE: example UpperMonomials; shows an example
"
{
  int i,d;
  intvec wt;
  ideal I, J;

  if(size(#) == 0) { wt = weight(f);}
  else { wt = #[1];}
   J = kbase(std(jacob(f)));
  d = deg(f, wt);
  for(i = 1; i <= size(J); i++) { if(deg(J[i], wt) > d) {I = I, J[i];} }
  return(simplify(I, 2));
}
example
{"EXAMPLE:";  echo = 2;
  ring B   = 0,(x,y,z), ls;
  poly f = -z5+y5+x2z+x2y;
  UpperMonomials(f);
}

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

proc ArnoldAction(poly f, list #)
"USAGE:   ArnoldAction(f, [Gf, B]); poly f; list Gf, B;
         'Gf' is a list of two rings (coming from 'StabEqn')
PURPOSE: compute the induced action of the stabilizer G of f on T_, where
         T_ is given by the upper monomials B of the Milnor algebra of f.
ASSUME:  f is quasihomogeneous
RETURN:  polynomial ring over the same groundfield, containing the ideals
         'actionid' and 'stabid'.
         - 'actionid' is the ideal defining the induced action of Gf on T_ @*
         - 'stabid' is the ideal of the stabilizer Gf in the new ring
EXAMPLE: example ArnoldAction; shows an example
"
{
  int i, offset, ub, pos, nrStabVars, dbPrt;
  intvec wt = weight(f);
  ideal B;
  list Gf, parts, baseDeg;
  string ringSTR1, ringSTR2, parName, namesSTR, varSTR;

  dbPrt = printlevel-voice+2;
  if(size(#) == 0) {
    Gf = StabEqn(f);
    B = UpperMonomials(f, wt);
  }
  else {
    Gf = #[1];
    if(size(#) > 1) { B = #[2];}
    else {B = UpperMonomials(f, wt);}
  }
  if(size(B) == 0) { ERROR("the principal part " + string(f) + " has no upper monomials");}
  for(i = 1; i <= size(B); i = i + 1) {
    baseDeg[i] = deg(B[i], wt);
  }
  ub = Max(baseDeg) + 1;          // max degree of an upper mono.
  def RAAB = basering;
  def STR1 = Gf[1];
  def STR2 = Gf[2];
  nrStabVars = nvars(STR1);

  dbprint(dbPrt, "ArnoldAction of f = ", f, ", upper base = " + string(B));

  setring STR1;
  poly mPoly = minpoly;
  setring RAAB;

  // setup new ring with s(..) and t(..) as parameters

  varSTR = string(maxideal(1));
  ringSTR2 = "ring RAAS = ";
  if(npars(basering) == 1) {
    parName = parstr(basering);
    ringSTR2 = ringSTR2 + "(0, " + parstr(1) + "), ";
  }
  else {
    parName = "a";
    ringSTR2 = ringSTR2 + "0, ";
  }
  offset = 1 + nrStabVars;
  namesSTR = "s(1.." + string(nrStabVars) + "), t(1.." + string(size(B)) + ")";
  ringSTR2 = ringSTR2 + "(" + namesSTR + "), lp;";
  ringSTR1 = "ring RAAR = (0, " + parName + "," + namesSTR + "), (" + varSTR + "), ls;";  // lp ?

  execute(ringSTR1);
  export(RAAR);
  ideal upperBasis, stabaction, action, mPoly, reduceIdeal;
  poly f, F, monos, h;

  reduceIdeal = imap(STR1, mPoly), imap(STR1, stabid);
  f = imap(RAAB, f);
  F = f;
  upperBasis = imap(RAAB, B);
  for(i = 1; i <= size(upperBasis); i = i + 1) {
    F = F + par(i + offset)*upperBasis[i];
  }
  monos = F - f;
  stabaction = imap(STR2, actionid);

  // action of the stabilizer on the semiuniversal unfolding of f

  F = f + APSubstitution(monos, stabaction, reduceIdeal, wt, ub, nrStabVars, size(upperBasis));

  // apply the theorem of Arnold

  h = ArnoldFormMain(f, upperBasis, F, reduceIdeal, nrStabVars, size(upperBasis)) - f;

  // extract the polynomials of the action of the stabilizer on T_

  parts = MonosAndTerms(h, wt, ub);
  for(i = 1; i <= size(parts[1]); i = i + 1) {
    pos = FirstEntryQHM(upperBasis, parts[1][i]);
    action[pos] = parts[2][i]/parts[1][i];
  }
  execute(ringSTR2);
  minpoly = number(imap(STR1, mPoly));
  ideal actionid = imap(RAAR, action);
  ideal stabid = imap(STR1, stabid);
  export(actionid);
  export(stabid);
  kill(RAAR);
dbprint(dbPrt, "
// 'ArnoldAction' created a new ring.
// To see the ring, type (if the name of the ring is R):
     show(R);
// To access the ideal of the stabilizer G of f and its group action,
// where f is the quasihomogeneous principal part, type
     def R = ArnoldAction(f); setring R;  stabid; actionid;
// 'stabid' is the ideal of the group G and 'actionid' is the ideal defining
// the group action of the group G on T_. Note: this action might be nonlinear
");
  return(RAAS);
}
example
{"EXAMPLE:";  echo = 2;
  ring B   = 0,(x,y,z), ls;
  poly f = -z5+y5+x2z+x2y;
  def R = ArnoldAction(f);
  setring R;
  actionid;
  stabid;
}

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

proc StabOrder(list #)
"USAGE:   StabOrder(f); poly f;
PURPOSE: compute the order of the stabilizer group of f.
ASSUME:  f quasihomogeneous polynomial with an isolated singularity at 0
RETURN:  int
GLOBAL: varSubsList
"
{
  list stab;

  if(size(#) == 1) { stab = StabEqn(#[1]); }
  else {  stab = #;}

  def RSTO = stab[1];
  setring(RSTO);
  return(vdim(std(stabid)));
}

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

proc StabEqn(poly f)
"USAGE:   StabEqn(f); f polynomial
PURPOSE: compute the equations of the isometry group of f.
ASSUME:  f semiquasihomogeneous polynomial with an isolated singularity at 0
RETURN:  list of two ring 'S1', 'S2'
         - 'S1' contians the equations of the stabilizer (ideal 'stabid') @*
         - 'S2' contains the action of the stabilizer (ideal 'actionid')
EXAMPLE: example StabEqn; shows an example
GLOBAL: varSubsList, contains the index j s.t. x(i) -> x(i)t(j) ...
"
{
dbprint(dbPrt, "
// 'StabEqn' created a list of 2 rings.
// To see the rings, type (if the name of your list is stab):
     show(stab);
// To access the 1-st ring and map (and similair for the others), type:
     def S1 = stab[1]; setring S1;  stabid;
// S1/stabid is the coordinate ring of the variety of the
// stabilizer, say G. If G x K^n --> K^n is the action of G on
// K^n, then the ideal 'actionid' in the second ring describes
// the dual map on the ring level.
// To access the 2-nd ring and map (and similair for the others), type:
     def S2 = stab[2]; setring S2;  actionid;
");

        return(StabEqnId(ideal(f), qhweight(f)));
}
example
{"EXAMPLE:";  echo = 2;
  ring B = 0,(x,y,z), ls;
  poly f = -z5+y5+x2z+x2y;
  list stab = StabEqn(f);
  def S1 = stab[1]; setring S1;  stabid;
  def S2 = stab[2]; setring S2;  actionid;
}

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

proc StabEqnId(ideal data, intvec wt)
"USAGE:   StabEqn(I, w); I ideal, w intvec
PURPOSE: compute the equations of the isometry group of the ideal I
         each generator of I is fixed by the stabilizer.
ASSUME:  I semiquasihomogeneous ideal wrt 'w' with an isolated singularity at 0
RETURN:  list of two ring 'S1', 'S2'
         - 'S1' contians the equations of the stabilizer (ideal 'stabid') @*
         - 'S2' contains the action of the stabilizer (ideal 'actionid')
EXAMPLE: example StabEqnId; shows an example
GLOBAL: varSubsList, contains the index j s.t. t(i) -> t(i)t(j) ...
"
{
  int i, j, c, k, r, nrVars, offset, n, sln, dbPrt;
  list variables, rd, temp, sList, varSubsList;
  poly mPoly;
  string ringSTR, ringSTR1, varString, parString;

  dbPrt = printlevel-voice+2;
  dbprint(dbPrt, "StabilizerEquations of " + string(data));

  export(varSubsList);
  n = nvars(basering);
  variables = StabVar(wt);    // possible quasihomogeneous substitutions
  nrVars = 0;
  for(i = 1; i <= size(wt); i = i + 1) {
    nrVars = nrVars + size(variables[i]);
  }

  // set the new basering needed for the substitutions

  varString = "s(1.." + string(nrVars) + ")";
  if(npars(basering) == 1) {
    parString = "(0, " + parstr(basering) + ")";
  }
  else { parString = "0"; }

  def RSTB = basering;
  mPoly = minpoly;
  ringSTR = "ring RSTR = " + parString + ", (" + varstr(basering) + ", " + varString + "), dp;";  // dp
        ringSTR1 = "ring RSTT = " + parString + ", (" + varString + ", " + varstr(basering) + "), dp;";

  if(defined(RSTR)) { kill(RSTR);}
        if(defined(RSTT)) { kill(RSTT);}
        execute(ringSTR1);      // this ring is only used for the result, where the variables
  export(RSTT);           // are s(1..m),t(1..n), as needed for Derksens algorithm (NullCone)
  minpoly = number(imap(RSTB, mPoly));

  execute(ringSTR);
  export(RSTR);
  minpoly = number(imap(RSTB, mPoly));
  poly f, f1, g, h, vars, pp;      // f1 is the polynomial after subs,
  ideal allEqns, qhsubs, actionid, stabid, J;
  list ringList;          // all t(i)`s which do not appear in f1
  ideal data = simplify(imap(RSTB, data), 2);

  // generate the quasihomogeneous substitution map F

  nrVars = 0;
  offset = 0;
  for(i = 1; i <= size(wt); i = i + 1) {    // build the substitution t(i) -> ...
    if(i > 1) { offset = offset + size(variables[i - 1]); }
    g = 0;
    for(j = 1; j <= size(variables[i]); j = j + 1) {
      pp = 1;
      for(k = 2; k <= size(variables[i][j]); k = k + 1) {
        pp = pp * var(variables[i][j][k]);
        if(variables[i][j][k] == i) { varSubsList[i] = offset + j;}
      }
      g = g + s(offset + j) * pp;
    }
    qhsubs[i] = g;
  }
  dbprint(dbPrt, "  qhasihomogenous substituion =" + string(qhsubs));
  map F = RSTR, qhsubs;
  kill(varSubsList);

  // get the equations of the stabilizer by comparing coefficients
  // in the equation f = F(f).

  vars = RingVarProduct(Table("i", "i", 1, size(wt)));

  allEqns = 0;

  for(r = 1; r <= ncols(data); r++) {

  f = data[r];
  f1 = F(f);
  int d = deg(f);
  matrix newcoMx = coef(f1, vars);        // coefficients of F(f)
  matrix coMx = coef(f, vars);          // coefficients of f

  for(i = 1; i <= ncols(newcoMx); i = i + 1) {      // build the system of eqns via coeff. comp.
    j = 1;
    h = 0;
    while(j <= ncols(coMx)) {        // all monomials in f
      if(coMx[j][1] == newcoMx[i][1]) { h = coMx[j][2]; j = ncols(coMx) + 1;}
      else {j = j + 1;}
    }
    J = J, newcoMx[i][2] - h;        // add equation
  }
  allEqns =  allEqns, J;

  }
  allEqns = std(allEqns);

  // simplify the equations, i.e., if s(i) in J then remove s(i) from J
  // and from the basering

  sList = SimplifyIdeal(allEqns, n, "s");
  stabid = sList[1];
  map phi = basering, sList[2];
        sln = size(sList[3]) - n;

  // change the substitution

  actionid = phi(qhsubs);

        // change to new ring, auxillary construction

        setring(RSTT);
        ideal actionid, stabid;

        actionid = imap(RSTR, actionid);
        stabid = imap(RSTR, stabid);
        export(stabid);
  export(actionid);
  ringList[2] = RSTT;

  dbprint(dbPrt, "  substitution = " + string(actionid));
  dbprint(dbPrt, "  equations of stabilizer = " + string(stabid));

  varString = "s(1.." + string(sln) + ")";
  ringSTR = "ring RSTS = " + parString + ", (" + varString + "), dp;";
  execute(ringSTR);
  minpoly = number(imap(RSTB, mPoly));
  ideal stabid = std(imap(RSTR, stabid));
  export(stabid);
  ringList[1] = RSTS;
dbprint(dbPrt, "
// 'StabEqnId' created a list of 2 rings.
// To see the rings, type (if the name of your list is stab):
     show(stab);
// To access the 1-st ring and map (and similair for the others), type:
     def S1 = stab[1]; setring S1;  stabid;
// S1/stabid is the coordinate ring of the variety of the
// stabilizer, say G. If G x K^n --> K^n is the action of G on
// K^n, then the ideal 'actionid' in the second ring describes
// the dual map on the ring level.
// To access the 2-nd ring and map (and similair for the others), type:
     def S2 = stab[2]; setring S2;  actionid;
");
  return(ringList);
}
example
{"EXAMPLE:";  echo = 2;
  ring B   = 0,(x,y,z), ls;
  ideal I = x2,y3,z6;
  intvec w = 3,2,1;
  list stab = StabEqnId(I, w);
  def S1 = stab[1]; setring S1;  stabid;
  def S2 = stab[2]; setring S2;  actionid;
}

///////////////////////////////////////////////////////////////////////////////
static
proc ArnoldFormMain(poly f, B, poly Fs, ideal reduceIdeal, int nrs, int nrt)
"USAGE:   ArnoldFormMain(f, B, Fs, rI, nrs, nrt);
   poly f,Fs; ideal B, rI; int nrs, nrt
PURPOSE: compute the induced action of 'G_f' on T_, where f is the principal
         part and 'Fs' is the semiuniversal unfolding of 'f' with x_i
         substituted by actionid[i], 'B' is a list of upper basis monomials
         for the milnor algebra of 'f', 'nrs' = number of variables for 'G_f'
         and 'nrt' = dimension of T_
ASSUME:  f is quasihomogeneous with an isolated singularity at 0,
         s(1..r), t(1..m) are parameters of the basering
RETURN:  poly
EXAMPLE: example ArnoldAction; shows an example
"
{
  int i, j, d, ub, dbPrt;
  list upperBasis, basisDegList, gmonos, common, parts;
  ideal jacobianId, jacobIdstd, mapId;    // needed for phi
  intvec wt = weight(f);
  matrix gCoeffMx;        // for lift command
  poly newFs, g, gred, tt;        // g = sum of all monomials of degree d, gred is needed for lift
  map phi;          // the map from Arnold's Theorem

  dbPrt = printlevel-voice+2;
  jacobianId = jacob(f);
  jacobIdstd = std(jacobianId);
  newFs = Fs;
  for(i = 1; i <= size(B); i = i + 1) {
    basisDegList[i] = deg(B[i], wt);
  }
  ub = Max(basisDegList) + 1;          // max degree of an upper monomial

  parts = MonosAndTerms(newFs - f, wt, ub);
  gmonos = parts[1];
  d = deg(f, wt);

  for(i = d + 1; i < ub; i = i + 1) {    // base[1] = monomials of degree i
    upperBasis[i] = SelectMonos(list(B, B), wt, i);    // B must not contain 0's
  }

  // test if each monomial of Fs is contained in B, if not,
  // compute a substitution via Arnold's theorem and substitutite
  // it into newFs

  for(i = d + 1; i < ub; i = i + 1) {  // ub instead of @UB
    dbprint(dbPrt, "-- degree = " + string(i) + " of " + string(ub - 1) + " ---------------------------");
    if(size(newFs) < 80) { dbprint(dbPrt, "  polynomial = " + string(newFs - f));}
    else {  dbprint(dbPrt, "  poly has deg (not weighted) " + string(deg(newFs)) + " and contains " + string(size(newFs)) + " monos");}

    // select monomials of degree i and intersect them with upperBasis[i]

    gmonos = SelectMonos(parts, wt, i);
    common = IntersectionQHM(upperBasis[i][1], gmonos[1]);
    if(size(common) == size(gmonos[1])) {
      dbprint(dbPrt, " no additional monomials ");
    }

    // other monomials than those in upperBasis occur, compute
    // the map constructed in the proof of Arnold's theorem
    // write g = c[i] * jacobianId[i]

    else {
      dbprint(dbPrt, "  additional Monomials found, compute the map ");
      g = PSum(gmonos[2]);      // sum of all monomials in g of degree i
      dbprint(dbPrt, "  sum of degree " + string(i) + " is " + string(g));

      gred = reduce(g, jacobIdstd);
      gCoeffMx = lift(jacobianId, g - gred);    // compute c[i]
      mapId = var(1) - gCoeffMx[1][1];    // generate the map
      for(j = 2; j <= size(gCoeffMx); j = j + 1) {
        mapId[j] = var(j) - gCoeffMx[1][j];
      }
      dbprint(dbPrt, "  map = " + string(mapId));
      // apply the map to newFs
      newFs = APSubstitution(newFs, mapId, reduceIdeal, wt, ub, nrs, nrt);
      parts = MonosAndTerms(newFs - f, wt, ub);  // monos and terms of deg < ub
      newFs = PSum(parts[2]) + f;      // result of APS... is already reduced
      dbprint(dbPrt, "  monomials of degree " + string(i));
    }
  }
  return(newFs);
}

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

static proc MonosAndTerms(poly f, wt, int ub)
"USAGE:   MonosAndTerms(f, w, ub); poly f, intvec w, int ub
PURPOSE: returns a list of all monomials and terms occuring in f of
         weighted degree < ub
RETURN:  list
         _[1]  list of monomials
         _[2]  list of terms
EXAMPLE: example MonosAndTerms shows an example
"
{
  int i, k;
  list monomials, terms;
  poly mono, lcInv, data;

  data = jet(f, ub - 1, wt);
  k = 0;
  for(i = 1; i <= size(data); i = i + 1) {
    mono = lead(data[i]);
    if(deg(mono, wt) < ub) {
      k = k + 1;
      lcInv = 1/leadcoef(mono);
      monomials[k] = mono * lcInv;
      terms[k] = mono;
    }
  }
  return(list(monomials, terms));
}
example
{"EXAMPLE:";  echo = 2;
  ring B = 0,(x,y,z), lp;
  poly f = 6*x2 + 2*x3 + 9*x*y2 + z*y + x*z6;
  MonosAndTerms(f, intvec(2,1,1), 5);
}

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

static proc SelectMonos(parts, intvec wt, int d)
"USAGE:   SelectMonos(parts, w, d); list/ideal parts, intvec w, int d
PURPOSE: returns a list of all monomials and terms occuring in f of
         weighted degree = d
RETURN:  list
         _[1]  list of monomials
         _[2]  list of terms
EXAMPLE: example SelectMonos; shows an example
"
{
  int i, k;
  list monomials, terms;
  poly mono;

  k = 0;
  for(i = 1; i <= size(parts[1]); i = i + 1) {
    mono = parts[1][i];
    if(deg(mono, wt) == d) {
      k = k + 1;
      monomials[k] = mono;
      terms[k] = parts[2][i];
    }
  }
  return(list(monomials, terms));
}
example
{"EXAMPLE:";  echo = 2;
  ring B = 0,(x,y,z), lp;
  poly f = 6*x2 + 2*x3 + 9*x*y2 + z*y + x*z6;
  list mt =  MonosAndTerms(f, intvec(2,1,1), 5);
  SelectMonos(mt, intvec(2,1,1), 4);
}

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

static proc Expand(substitution, degVec, ideal reduceI, intvec w1, int ub, list truncated)
"USAGE:   Expand(substitution, degVec, reduceI, w, ub, truncated);
         ideal/list substitution, list/intvec degVec, ideal reduceI, intvec w,
         int ub, list truncated
PURPOSE: substitute 'substitution' in the monomial given by the list of
         exponents 'degVec', omit all terms of weighted degree > ub and reduce
         the result w.r.t. 'reduceI'. If truncated[i] = 0 then the result is
         stored for later use.
RETURN:  poly
NOTE:    used by APSubstitution
GLOBAL:  computedPowers
"
{
  int i, minDeg;
  list powerList;
  poly g, h;

  // compute substitution[1]^degVec[1],...,subs[n]^degVec[n]

  for(i = 1; i <= ncols(substitution); i = i + 1) {
    if(size(substitution[i]) < 3 || degVec[i] < 4) {
      powerList[i] = reduce(substitution[i]^degVec[i], reduceI); // new
    }  // directly for small exponents
    else {
      powerList[i] = PolyPower1(i, substitution[i], degVec[i], reduceI, w1, truncated[i], ub);
    }
  }
  // multiply the terms obtained by using PolyProduct();
  g = powerList[1];
  minDeg = w1[1] * degVec[1];
  for(i = 2; i <= ncols(substitution); i = i + 1) {
    g = jet(g, ub - w1[i] * degVec[i] - 1, w1);
    h = jet(powerList[i], ub - minDeg - 1, w1);
    g = PolyProduct(g, h, reduceI, w1, ub);
    if(g == 0) { Print(" g = 0 "); break;}
    minDeg = minDeg + w1[i] * degVec[i];
  }
  return(g);
}

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

static proc PolyProduct(poly g1, poly h1, ideal reduceI, intvec wt, int ub)
"USAGE:   PolyProduct(g, h, reduceI, wt, ub); poly g, h; ideal reduceI,
          intvec wt, int ub.
PURPOSE: compute g*h and reduce it w.r.t 'reduceI' and omit terms of weighted
         degree > ub.
RETURN:  poly
NOTE:    used by 'Expand'
"
{
  int SUBSMAXSIZE = 3000;    //
  int i, nrParts, sizeOfPart, currentPos, partSize, maxSIZE;
  poly g, h, gxh, prodComp, @g2;    // replace @g2 by subst.

  g = g1;
  h = h1;

  if(size(g)*size(h) > SUBSMAXSIZE) {

    // divide the polynomials with more terms in parts s.t.
    // the product of each part with the other polynomial
    // has at most SUBMAXSIZE terms

    if(size(g) < size(h)) { poly @h = h; h = g; g = @h;@h = 0; }
    maxSIZE = SUBSMAXSIZE / size(h);
    //print(" SUBSMAXSIZE = "+string(SUBSMAXSIZE)+" exceeded by "+string(size(g)*size(h)) + ", maxSIZE = ", string(maxSIZE));
    nrParts = size(g) / maxSIZE + 1;
    partSize = size(g) / nrParts;
    gxh = 0;  // 'g times h'
    for(i = 1; i <= nrParts; i = i + 1){
      //print(" loop #" + string(i) + " of " + string(nrParts));
      currentPos = (i - 1) * partSize;
      if(i < nrParts) {sizeOfPart = partSize;}
      else { sizeOfPart = size(g) - (nrParts - 1) * partSize; print(" last #" + string(sizeOfPart) + " terms ");}
      prodComp = g[currentPos + 1..sizeOfPart + currentPos] * h;  // multiply a part
      @g2 = jet(prodComp, ub - 1, wt);  // eventual reduce ...
      if(size(@g2) < size(prodComp)) { print(" killed " + string(size(prodComp) - size(@g2)) + " terms ");}
      gxh =  reduce(gxh + @g2, reduceI);

    }
  }
  else {
    gxh = reduce(jet(g * h,ub - 1, wt), reduceI);
  }  // compute directly
  return(gxh);
}

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

static proc PolyPower1(int varIndex, poly f, int e, ideal reduceI, intvec wt, 
                       int truncated, int ub)
"USAGE:   PolyPower1(i, f, e, reduceI, wt, truncated, ub);int i, e, ub;poly f;
         ideal reduceI; intvec wt; list truncated;
PURPOSE: compute f^e, use previous computations if possible, and reduce it
         w.r.t reudecI and omit terms of weighted degree > ub.
RETURN:  poly
NOTE:    used by 'Expand'
GLOBAL:  'computedPowers'
"
{
  int i, ordOfg, lb, maxPrecomputedPower;
  poly g, fn;

  if(e == 0) { return(1);}
  if(e == 1) { return(f);}
  if(f == 0) { return(1); }
  else {

    // test if f has been computed to some power

    if(computedPowers[varIndex][1] > 0) {
      maxPrecomputedPower = computedPowers[varIndex][1];
      if(maxPrecomputedPower >= e) {
        // no computation necessary, f^e has already benn computed
        g = computedPowers[varIndex][2][e - 1];
        //Print("No computation, from list : g = elem [", varIndex, ", 2, ", e - 1, "]");
        lb = e + 1;
      }
      else {  // f^d computed, where d < e
        g = computedPowers[varIndex][2][maxPrecomputedPower - 1];
        ordOfg = maxPrecomputedPower * wt[varIndex];
        lb = maxPrecomputedPower + 1;
      }
    }
    else {    // no precomputed data
      lb = 2;
      ordOfg = wt[varIndex];
      g = f;
    }
    for(i = lb; i <= e; i = i + 1) {
      fn = jet(f, ub - ordOfg - 1, wt); // reduce w.r.t. reduceI
      g = PolyProduct(g, fn, reduceI, wt, ub);
      ordOfg = ordOfg + wt[varIndex];
      if(g == 0) { break; }
      if((i > maxPrecomputedPower) && !truncated) {
        if(maxPrecomputedPower == 0) {  // init computedPowers
          computedPowers[varIndex] = list(i, list(g));
        }
        computedPowers[varIndex][1] = i;  // new degree
        computedPowers[varIndex][2][i - 1] = g;
        maxPrecomputedPower = i;
      }
    }
  }
  return(g);
}

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

static proc RingVarsToList(list @index)
{
  int i;
  list temp;

  for(i = 1; i <= size(@index); i = i + 1) { temp[i] = string(var(@index[i])); }
  return(temp);
}

///////////////////////////////////////////////////////////////////////////////
static
proc APSubstitution(poly f, ideal substitution, ideal reduceIdeal, intvec wt, int ub, int nrs, int nrt)
"USAGE:   APSubstitution(f, subs, reduceI, w, ub, int nrs, int nrt); poly f
         ideal subs, reduceI, intvec w, int ub, nrs, nrt;
         nrs = number of parameters s(1..nrs),
         nrt = number of parameters t(1..nrt)
PURPOSE: substitute 'subs' in f, omit all terms with weighted degree > ub and
         reduce the result w.r.t. 'reduceI'.
RETURN:  poly
GLOBAL:  'computedPowers'
"
{
  int i, j, k, d, offset;
  int n = nvars(basering);
  list  coeffList, parts, degVecList, degOfMonos;
  list computedPowers, truncatedQ, degOfSubs, @temp;
  string ringSTR, @ringVars;

  export(computedPowers);

  // store arguments in strings

  def RASB = basering;

  parts = MonosAndTerms(f, wt, ub);
  for(i = 1; i <= size(parts[1]); i = i + 1) {
    coeffList[i] = parts[2][i]/parts[1][i];
    degVecList[i] = leadexp(parts[1][i]);
    degOfMonos[i] = deg(parts[1][i], wt);
  }

  // built new basering with no parameters and order dp !
  // the parameters of the basering are appended to
  // the variables of the basering !
  // set ideal mpoly = minpoly for reduction !

  @ringVars = "(" + varstr(basering) + ", " + parstr(1) + ",";  // precondition
  if(nrs > 0) {
    @ringVars = @ringVars + "s(1.." + string(nrs) + "), ";
  }
  @ringVars = @ringVars + "t(1.." + string(nrt) + "))";
  ringSTR = "ring RASR = 0, " + @ringVars + ", dp;";    // new basering

  // built the "reduction" ring with the reduction ideal

  execute(ringSTR);
  export(RASR);
  ideal reduceIdeal, substitution, newSubs;
  intvec w1, degVec;
  list minDeg, coeffList, degList;
  poly f, g, h, subsPoly;

  w1 = wt;            // new weights
  offset = nrs + nrt + 1;
  for(i = n + 1; i <= offset + n; i = i + 1) { w1[i] = 0; }

  reduceIdeal = std(imap(RASB, reduceIdeal)); // omit later !
  coeffList = imap(RASB, coeffList);
  substitution = imap(RASB, substitution);

  f = imap(RASB, f);

  for(i = 1; i <= n; i = i + 1) {      // all "base" variables
    computedPowers[i] = list(0);
    for(j = 1; j <= size(substitution[i]); j = j + 1) { degList[j] = deg(substitution[i][j], w1);}
    degOfSubs[i] = degList;
  }

  // substitute in each monomial seperately

  g = 0;
  for(i = 1; i <= size(degVecList); i = i + 1) {
    truncatedQ = Table("0", "i", 1, n);
    newSubs = 0;
    degVec = degVecList[i];
    d = degOfMonos[i];

    // check if some terms in the substitution can be omitted
    // degVec = list of exponents of the monomial m
    // minDeg[j] denotes the weighted degree of the monomial m'
    // where m' is the monomial m without the j-th variable

    for(j = 1; j <= size(degVec); j = j + 1) { minDeg[j] = d - degVec[j] * wt[j]; }

    for(j = 1; j <= size(degVec); j = j + 1) {
      subsPoly = 0;      // set substitution to 0
      if(degVec[j] > 0) {

        // if variable occurs then check if
        // substitution[j][k] * (linear part)^(degVec[j]-1) + minDeg[j] < ub
        // i.e. look for the smallest possible combination in subs[j]^degVec[j]
        // which comes from the term substitution[j][k]. This term is multiplied
        // with the rest of the monomial, which has at least degree minDeg[j].
        // If the degree of this product is < ub then substitution[j][k] contributes
        // to the result and cannot be omitted

        for(k = 1; k <= size(substitution[j]); k = k + 1) {
          if(degOfSubs[j][k] + (degVec[j] - 1) * wt[j] + minDeg[j]  < ub) {
            subsPoly = subsPoly + substitution[j][k];
          }
        }
      }
      newSubs[j] = subsPoly;        // set substitution
      if(substitution[j] - subsPoly != 0) { truncatedQ[j] = 1;}  // mark that substitution[j] is truncated
    }
    h = Expand(newSubs, degVec, reduceIdeal, w1, ub, truncatedQ) * coeffList[i];  // already reduced
    g = reduce(g + h, reduceIdeal);
  }
   kill(computedPowers);
  setring RASB;
  poly fnew = imap(RASR, g);
  kill(RASR);
  return(fnew);
}

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

static proc StabVar(intvec wt)
"USAGE:   StabVar(w); intvec w
PURPOSE: compute the indicies for quasihomogeneous substitutions of each
         variable.
ASSUME:  f semiquasihomogeneous polynomial with an isolated singularity at 0
RETURN:  list
         _[i]  list of combinations for var(i) (i must be appended
         to each comb)
GLOBAL:  'varSubsList', contains the index j s.t. x(i) -> x(i)t(j) ...
"
{
  int i, j, k, uw, ic;
  list varList, variables, subs;
  string str, varString;

  varList = StabVarComb(wt);
  for(i = 1; i <= size(wt); i = i + 1) {
    subs = 0;

    // built linear substituitons
    for(j = 1; j <= size(varList[1][i]); j = j + 1) {
      subs[j] = list(i) + list(varList[1][i][j]);
    }
    variables[i] = subs;
    if(size(varList[2][i]) > 0) {

      // built nonlinear substituitons
      subs = 0;
      for(j = 1; j <= size(varList[2][i]); j = j + 1) {
        subs[j] = list(i) + varList[2][i][j];
      }
      variables[i] = variables[i] + subs;
    }
  }
  return(variables);
}

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

static proc StabVarComb(intvec wt)
"USAGE:   StabVarComb(w); intvec w
PURPOSE: list all possible indices of indeterminates for a quasihom. subs.
RETURN:  list
         _[1] linear substitutions
         _[2] nonlinear substiutions
GLOBAL: 'varSubsList', contains the index j s.t. x(i) -> x(i)t(j) ...
"
{
  int mi, ma, i, j, k, uw, ic;
  list index, indices, usedWeights, combList, combinations;
  list linearSubs, nonlinearSubs;
  list partitions, subs, temp;        // subs[i] = substitution for var(i)

  linearSubs = Table("0", "i", 1, size(wt));
  nonlinearSubs = Table("0", "i", 1, size(wt));

  uw = 0;
  ic = 0;
  mi = Min(wt);
  ma = Max(wt);

  for(i = mi; i <= ma; i = i + 1) {
    if(ContainedQ(wt, i)) {    // find variables of weight i
      k = 0;
      index = 0;
      // collect the indices of all variables of weight i
      for(j = 1; j <= size(wt); j = j + 1) {
        if(wt[j] == i) {
          k = k + 1;
          index[k] = j;
        }
      }
      uw = uw + 1;
      usedWeights[uw] = i;
      ic = ic + 1;
      indices[i] = index;

      // linear part of the substitution

      for(j = 1; j <= size(index); j = j + 1) {
        linearSubs[index[j]] = index;
      }

      // nonlinear part of the substitution

      if(uw > 1) {    // variables of least weight do not allow nonlinear subs.

      partitions = Partitions(i, delete(usedWeights, uw));
      for(j = 1; j <= size(partitions); j = j + 1) {
        combinations[j] = AllCombinations(partitions[j], indices);
      }
      for(j = 1; j <= size(index); j = j + 1) {
        nonlinearSubs[index[j]] = FlattenQHM(combinations);   // flatten one level !
      }

      } // end if

    }
  }
  combList[1] = linearSubs;
  combList[2] = nonlinearSubs;
  return(combList);
}

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

static proc AllCombinations(list partition, list indices)
"USAGE:   AllCombinations(partition,indices); list partition, indices)
PURPOSE: all combinations for a given partititon
RETURN:  list
GLOBAL: varSubsList, contains the index j s.t. x(i) -> x(i)t(j) ...
"
{
  int i, k, m, ok, p, offset;
  list nrList, indexList;

  k = 0;
  offset = 0;
  i = 1;
  ok = 1;
  m = partition[1];
  while(ok) {
    if(i > size(partition)) {
      ok = 0;
      p = 0;
    }
    else { p = partition[i];}
    if(p == m) { i = i + 1;}
    else {
      k = k + 1;
      nrList[k] = i - 1 - offset;
      offset = offset + i - 1;
      indexList[k] = indices[m];
      if(ok) { m = partition[i];}
    }
  }
  return(AllCombinationsAux(nrList, indexList));
}

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

static proc AllSingleCombinations(int n, list index)
"USAGE:   AllSingleCombinations(n index); int n, list index
PURPOSE: all combinations for var(n)
RETURN:  list
"
{
  int i, j, k;
  list comb, newC, temp, newIndex;

  if(n == 1) {
    for(i = 1; i <= size(index); i = i + 1) {
      temp = index[i];
      comb[i] = temp;
    }
    return(comb);
  }
  if(size(index) == 1) {
    temp = Table(string(index[1]), "i", 1, n);
    comb[1] = temp;
    return(comb);
  }
  newIndex = index;
  for(i = 1; i <= size(index); i = i + 1) {
    if(i > 1) { newIndex = delete(newIndex, 1); }
    newC = AllSingleCombinations(n - 1, newIndex);
    k = size(comb);
    temp = 0;
    for(j = 1; j <= size(newC); j = j + 1) {
      temp[1] = index[i];
      temp = temp + newC[j];
      comb[k + j] = temp;
      temp = 0;
    }
  }
  return(comb);
}

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

static proc AllCombinationsAux(list parts, list index)
"USAGE:  AllCombinationsAux(parts ,index); list parts, index
PURPOSE: all compbinations for nonlinear substituiton
RETURN:  list
"
{
  int i, j, k;
  list comb, firstC, restC;

  if(size(parts) == 0 || size(index) == 0) { return(comb);}

  firstC = AllSingleCombinations(parts[1], index[1]);
  restC = AllCombinationsAux(delete(parts, 1), delete(index, 1));

  if(size(restC) == 0) { comb = firstC;}
  else {
    for(i = 1; i <= size(firstC); i = i + 1) {
      k = size(comb);
      for(j = 1; j <= size(restC); j = j + 1) {
        //elem = firstC[i] + restC[j];
        // comb[k + j] = elem;
        comb[k + j] = firstC[i] + restC[j];
      }
    }
  }
  return(comb);
}

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

static proc Partitions(int n, list nr)
"USAGE:   Partitions(n, nr); int n, list nr
PURPOSE: partitions of n consisting of elements from nr
RETURN:  list
"
{
  int i, j, k;
  list parts, temp, restP, newP, decP;

  if(size(nr) == 0) { return(list());}
  if(size(nr) == 1) {
    if(NumFactor(nr[1], n) > 0) {
      parts[1] = Table(string(nr[1]), "i", 1, NumFactor(nr[1], n));
    }
    return(parts);
  }
  else {
    parts =  Partitions(n, nr[1]);
    restP = Partitions(n, delete(nr, 1));

    parts = parts + restP;
    for(i = 1; i <= n / nr[1]; i = i + 1) {
      temp = Table(string(nr[1]), "i", 1, i);
      decP = Partitions(n - i*nr[1], delete(nr, 1));
      k = size(parts);
      for(j = 1; j <= size(decP); j = j + 1) {
        newP = temp + decP[j];        // new partition
        if(!ContainedQ(parts, newP, 1)) {
          k = k + 1;
          parts[k] = newP;
        }
      }
    }
  }
  return(parts);
}

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

static proc NumFactor(int a, int b)
" USAGE: NumFactor(a, b); int a, b
PURPOSE: if b divides a then return b/a, else return 0
RETURN:  int
"
{
  int c = int(b/a);
  if(c*a == b) { return(c); }
  else {return(0)}
}

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

static proc Table(string cmd, string iterator, int lb, int ub)
" USAGE: Table(cmd,i, lb, ub); string cmd, i; int lb, ub
PURPOSE: generate a list of size ub - lb + 1 s.t. _[i] = cmd(i)
RETURN:  list
"
{
  list data;
  execute("int " + iterator + ";");

  for(int @i = lb; @i <= ub; @i++) {
    execute(iterator + " = " + string(@i));
    execute("data[" + string(@i) + "] = " + cmd + ";");
  }
  return(data);
}

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

static proc FlattenQHM(list data)
" USAGE: FlattenQHM(n, nr); list data
PURPOSE: flatten the list (one level) 'data', which is a list of lists
RETURN:  list
"
{
  int i, j, c;
  list fList, temp;

  c = 1;

  for(i = 1; i <= size(data); i = i + 1) {
    for(j = 1; j <= size(data[i]); j = j + 1) {
      fList[c] = data[i][j];
      c = c + 1;
    }
  }
  return(fList);
}

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

static proc IntersectionQHM(list l1, list l2)
// Type : list
// Purpose : Intersection of l1 and l2
{
  list l;
  int b, c;

  c = 1;

  for(int i = 1; i <= size(l1); i = i + 1) {
    b = ContainedQ(l2, l1[i]);
    if(b == 1) {
      l[c] = l1[i];
      c = c + 1;
    }
  }
  return(l);
}

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

static proc FirstEntryQHM(data, elem)
// Type : int
// Purpose : position of first entry equal to elem in data (from left to right)
{
  int i, pos;

  i = 0;
  pos = 0;
  while(!pos && i < size(data)) {
    i = i + 1;
    if(data[i] == elem) { pos = i;}
  }
  return(pos);
}

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

static proc PSum(e)
{
  poly f;
  for(int i = 1; i <= size(e); i = i + 1) {
    f = f + e[i];
  }
  return(f);

}

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

proc Max(data)
"USAGE:   Max(data); intvec/list of integers data
PURPOSE: find the maximal integer contained in 'data'
RETURN:  list
ASSUME:  'data' contians only integers and is not empty
"
{
  int i;
  int max = data[1];

  for(i = 1; i <= size(data); i = i + 1) {
    if(data[i] > max) { max = data[i]; }
  }
  return(max);
}

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

proc Min(data)
"USAGE:   Min(data); intvec/list of integers data
PURPOSE: find the minimal integer contained in 'data'
RETURN:  list
ASSUME:  'data' contians only integers and is not empty
"
{
  int i;
  int min = data[1];

  for(i = 1; i <= size(data); i = i + 1) {
    if(data[i] < min) { min = data[i]; }
  }
  return(min);
}

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