source: git/Singular/LIB/mondromy.lib @ 1eb7af

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

version="$Id: mondromy.lib,v 1.2 1999-06-22 14:50:49 mschulze Exp $";
info="
LIBRARY: mondromy.lib  PROCEDURES TO COMPUTE THE MONODROMY OF A SINGULARITY
AUTHOR:  Mathias Schulze, email: mschulze@mathematik.uni-kl.de

 invunit(u,n);         series inverse of polynomial u up to order n
 detadj(U);            determinant and adjoint matrix of square matrix U
 jacoblift(f);         lifts f^kappa in jacob(f) with minimal kappa
 monodromy(f[,opt]);   monodromy of isolated hypersurface singularity f
 H''basis(f);          basis of Brieskorn lattice H''
";

LIB "ring.lib";
LIB "sing.lib";
LIB "jordan.lib";
///////////////////////////////////////////////////////////////////////////////

static proc pcvladdl(list l1,list l2)
{
  return(system("pcvLAddL",l1,l2));
}

static proc pcvpmull(poly p,list l)
{
  return(system("pcvPMulL",p,l));
}

static proc pcvmindeg(list #)
{
  return(system("pcvMinDeg",#[1]));
}

static proc pcvp2cv(list l,int i0,int i1)
{
  return(system("pcvP2CV",l,i0,i1));
}

static proc pcvcv2p(list l,int i0,int i1)
{
  return(system("pcvCV2P",l,i0,i1));
}

static proc pcvdim(int i0,int i1)
{
  return(system("pcvDim",i0,i1));
}

static proc pcvbasis(int i0,int i1)
{
  return(system("pcvBasis",i0,i1));
}

static proc pcvinit()
{
  if(system("with","DynamicLoading"))
  {
    pcvlal=Pcv::LAddL;
    pcvpmull=Pcv::PMulL;
    pcvmindeg=Pcv::MinDeg;
    pcvp2cv=Pcv::P2CV;
    pcvcv2p=Pcv::CV2P;
    pcvdim=Pcv::Dim;
    pcvbasis=Pcv::Basis;
  }
}

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

static proc min(intvec v)
{
  int m=v[1];
  int i;
  for(i=2;i<=size(v);i++)
  {
    if(m>v[i])
    {
      m=v[i];
    }
  }
  return(m);
}
///////////////////////////////////////////////////////////////////////////////

static proc max(intvec v)
{
  int m=v[1];
  int i;
  for(i=2;i<=size(v);i++)
  {
    if(m<v[i])
    {
      m=v[i];
    }
  }
  return(m);
}
///////////////////////////////////////////////////////////////////////////////

static proc mdivp(matrix m,poly p)
{
  int i,j;
  for(i=nrows(m);i>=1;i--)
  {
    for(j=ncols(m);j>=1;j--)
    {
      m[i,j]=m[i,j]/p;
    }
  }
  return(m);  
}
///////////////////////////////////////////////////////////////////////////////

proc codimV(list V,int N)
{
  int codim=pcvdim(0,N);
  if(size(V)>0)
  {
    dbprint(printlevel-voice+2,"//vector space dimension: "+string(codim));
    dbprint(printlevel-voice+2,
      "//number of subspace generators: "+string(size(V)));
    int t=timer;
    codim=codim-ncols(interred(module(V[1..size(V)])));
    dbprint(printlevel-voice+2,"//codimension: "+string(codim));
  }
  return(codim);
}
///////////////////////////////////////////////////////////////////////////////

proc quotV(list V,int N)
{
  module Q=freemodule(pcvdim(0,N));
  if(size(V)>0)
  {
    dbprint(printlevel-voice+2,"//vector space dimension: "+string(nrows(Q)));
    dbprint(printlevel-voice+2,
      "//number of subspace generators: "+string(size(V)));
    int t=timer;
    Q=interred(reduce(std(Q),std(module(V[1..size(V)]))));
  }
  return(list(Q[1..size(Q)]));
}
///////////////////////////////////////////////////////////////////////////////

proc invunit(poly u,int n)
"USAGE:   invunit(u,n); u poly, n int
ASSUME:  The polynomial u is a series unit.
RETURN:  The procedure returns the series inverse of u up to order n
         or a zero polynomial if u is no series unit.
DISPLAY: printlevel>=1; shows comments.
EXAMPLE: example invunit; shows an example.
"
{
  if(pcvmindeg(u)==0)
  {
    def br=basering;
    changeord("pr","dp");
    qring qr=std(maxideal(n+1));
    kill pr;

    dbprint(printlevel-voice+2,"//computing inverse...");
    int t=timer;
    poly v=lift(fetch(br,u),1)[1,1];
    dbprint(printlevel-voice+2,"//...inverse computed ["+string(timer-t)+
      " secs]");

    setring br;
    return(fetch(qr,v));
  }
  else
  {
    print("//no series unit");
    return(poly(0));
  }
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0,x,dp;
  invunit(1+x,10);
}
///////////////////////////////////////////////////////////////////////////////

proc detadj(module U)
"USAGE:   detadj(U); U matrix
ASSUME:  U is a square matrix with non zero determinant.
RETURN:  The procedure returns a list with at most 2 entries.
         If U is not a sqaure matrix, the list is empty.
         If U is a sqaure matrix, then the first entry is the determinant of U.
         If U is a square matrix and the determinant of U not zero, 
         then the second entry is the adjoint matrix of U.
DISPLAY: printlevel>=1; shows comments.
EXAMPLE: example detadj; shows an example.
"
{
  if(nrows(U)==ncols(U))
  {
    dbprint(printlevel-voice+2,"//computing determinant...");
    int t=timer;
    poly detU=det(U);
    dbprint(printlevel-voice+2,"//...determinant computed ["+string(timer-t)+
      " secs]");

    if(detU==0)
    {
      print("//determinant zero");
      return(list(detU));
    }
    else
    {
      def br=basering;
      changeord("pr","dp");
      matrix U=fetch(br,U);
      poly detU=fetch(br,detU);

      dbprint(printlevel-voice+2,"//computing adjoint matrix...");
      t=timer;
      matrix adjU=lift(U,detU*freemodule(nrows(U)));
      dbprint(printlevel-voice+2,"//...adjoint matrix computed ["
        +string(timer-t)+" secs]");

      setring br;
      matrix adjU=fetch(pr,adjU);
      kill pr;
    }
  }
  else
  {
    print("//no square matrix");
    return(list());
  }

  return(list(detU,adjU));
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0,x,dp;
  matrix U[2][2]=1,1+x,1+x2,1+x3;
  list daU=detadj(U);
  daU[1];
  print(daU[2]);
}
///////////////////////////////////////////////////////////////////////////////

proc jacoblift(poly f)
"USAGE:   jacoblift(f); f poly
ASSUME:  The polynomial f in a series ring (local ordering) defines 
         an isolated hypersurface singularity.
RETURN:  The procedure returns a list with entries kappa, xi, u of type 
         int, vector, poly such that kappa is minimal with f^kappa in jacob(f),
         u is a unit, and u*f^kappa=(matrix(jacob(f))*xi)[1,1].
DISPLAY: printlevel>=1; shows comments.
EXAMPLE: example jacoblift; shows an example.
"
{
  dbprint(printlevel-voice+2,"//computing kappa...");
  int t=timer;
  ideal jf=jacob(f);
  ideal sjf=std(jf);
  int kappa=1;
  poly fkappa=f;
  while(reduce(fkappa,sjf)!=0)
  {
    dbprint(printlevel-voice+2,"//kappa="+string(kappa));
    kappa++;
    fkappa=fkappa*f;
  }
  dbprint(printlevel-voice+2,"//kappa="+string(kappa));
  dbprint(printlevel-voice+2,"//...kappa computed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,"//computing xi...");
  t=timer;
  vector xi=lift(jf,fkappa)[1];
  dbprint(printlevel-voice+2,"//...xi computed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,"//computing u...");
  t=timer;
  poly u=(matrix(jf)*xi)[1,1]/fkappa;
  dbprint(printlevel-voice+2,"//...u computed ["+string(timer-t)+" secs]");

  return(list(kappa,xi,u));
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0,(x,y),ds;
  poly f=x2y2+x6+y6;
  jacoblift(f);
}
///////////////////////////////////////////////////////////////////////////////

static proc getdeltaP1(poly f,int K,int N,int dN)
{
  return(pcvpmull(f^K,pcvbasis(0,N+dN-K*pcvmindeg(f))));
}
///////////////////////////////////////////////////////////////////////////////

static proc getdeltaP2(poly f,int N,int dN)
{
  def of,jf=pcvmindeg(f),jacob(f);
  list b=pcvbasis(N-of+2,N+dN-of+2);
  list P2;
  P2[size(b)*((nvars(basering)-1)*nvars(basering))/2]=0;
  int i,j,k,l;
  intvec alpha;
  for(k,l=1,1;k<=size(b);k++)
  {
    alpha=leadexp(b[k]);
    for(i=nvars(basering)-1;i>=1;i--)
    {
      for(j=nvars(basering);j>i;j--)
      {
        P2[l]=alpha[i]*jf[j]*(b[k]/var(i))-alpha[j]*jf[i]*(b[k]/var(j));
        l++;
      }
    }
  }
  return(P2);
}
///////////////////////////////////////////////////////////////////////////////

static proc getdeltaPe(poly f,list e,int K,int dK)
{
  int k;
  list Pe,fke;
  for(k,fke=K,pcvpmull(f^K,e);k<K+dK;k,fke=k+1,pcvpmull(f,fke))
  {
    Pe=Pe+fke;
  }
  return(Pe);
}
///////////////////////////////////////////////////////////////////////////////

static proc incK(poly f,int mu,int K,int deltaK,int N,
  list e,list P1,list P2,list Pe,list V1,list V2,list Ve)
{
  int deltaN=deltaK*pcvmindeg(f);

  list deltaP1;
  P1=pcvpmull(f^deltaK,P1);
  V1=pcvp2cv(P1,0,N+deltaN);

  list deltaP2=getdeltaP2(f,N,deltaN);
  V2=pcvladdl(V2,pcvp2cv(P2,N,N+deltaN))+pcvp2cv(deltaP2,0,N+deltaN);
  P2=P2+deltaP2;

  list deltaPe=getdeltaPe(f,e,K,deltaK);
  Ve=pcvladdl(Ve,pcvp2cv(Pe,N,N+deltaN))+pcvp2cv(deltaPe,0,N+deltaN);
  Pe=Pe+deltaPe;

  K=K+deltaK;
  dbprint(printlevel-voice+2,"//K="+string(K));

  N=N+deltaN;
  dbprint(printlevel-voice+2,"//N="+string(N));

  deltaN=1;
  dbprint(printlevel-voice+2,"//computing codimension of");
  dbprint(printlevel-voice+2,"//df^dOmega^(n-1)+f^K*Omega^(n+1) in "
    +"Omega^(n+1) mod m^N*Omega^(n+1)...");
  int t=timer;
  while(codimV(V1+V2,N)<K*mu)
  {
    dbprint(printlevel-voice+2,"//...codimension computed ["+string(timer-t)
      +" secs]");

    deltaP1=getdeltaP1(f,K,N,deltaN);
    V1=pcvladdl(V1,pcvp2cv(P1,N,N+deltaN))+pcvp2cv(deltaP1,0,N+deltaN);
    P1=P1+deltaP1;

    deltaP2=getdeltaP2(f,N,deltaN);
    V2=pcvladdl(V2,pcvp2cv(P2,N,N+deltaN))+pcvp2cv(deltaP2,0,N+deltaN);
    P2=P2+deltaP2;

    Ve=pcvladdl(Ve,pcvp2cv(Pe,N,N+deltaN));

    N=N+deltaN;
    dbprint(printlevel-voice+2,"//N="+string(N));

    dbprint(printlevel-voice+2,"//computing codimension of");
    dbprint(printlevel-voice+2,"//df^dOmega^(n-1)+f^K*Omega^(n+1) in "
      +"Omega^(n+1) mod m^N*Omega^(n+1)...");
    t=timer;
  }
  dbprint(printlevel-voice+2,"//...codimension computed ["+string(timer-t)
    +" secs]");

  return(K,N,P1,P2,Pe,V1,V2,Ve);
}
///////////////////////////////////////////////////////////////////////////////

static proc nablaK(poly f,int kappa,vector xi,poly u,int N,int prevN,
  list Vnablae,list e)
{
  xi=jet(xi,N);
  u=invunit(u,N);
  poly fkappa=kappa*f^(kappa-1);

  poly p,q;
  list nablae;
  int i,j;
  for(i=1;i<=size(e);i++)
  {
    for(j,p=nvars(basering),0;j>=1;j--) 
    {
      q=jet(e[i]*xi[j],N);
      if(q!=0)
      {
        p=p+diff(q*jet(u,N-pcvmindeg(q)),var(j));
      }
    }
    nablae=nablae+list(p-jet(fkappa*e[i],N-1));
  }

  return(pcvladdl(Vnablae,pcvp2cv(nablae,prevN,N-prevN)));
}
///////////////////////////////////////////////////////////////////////////////

static proc MK(poly f,int mu,int kappa,vector xi,poly u,
  int K,int N,int prevN,list e,list V1,list V2,list Ve,list Vnablae)
{
  dbprint(printlevel-voice+2,"//computing nabla(e)...");
  int t=timer;
  Vnablae=nablaK(f,kappa,xi,u,N,prevN,Vnablae,e);
  dbprint(printlevel-voice+2,"//...nabla(e) computed ["+string(timer-t)
    +" secs]");

  dbprint(printlevel-voice+2,
    "//lifting nabla(e) to C-basis of H''/t^KH''...");
  list V=Ve+V1+V2;
  module W=module(V[1..size(V)]);
  dbprint(printlevel-voice+2,"//vector space dimension: "+string(nrows(W)));
  dbprint(printlevel-voice+2,"//number of generators: "+string(ncols(W)));
  t=timer;
  matrix C=lift(W,module(Vnablae[1..size(Vnablae)]));
  dbprint(printlevel-voice+2,"//...nabla(e) lifted ["+string(timer-t)
    +" secs]");

  dbprint(printlevel-voice+2,"//computing e-lift of nabla(e)...");
  t=timer;
  int i1,i2,j,k;
  matrix M[mu][mu];
  for(j=1;j<=mu;j++)
  {
    for(k,i2=0,1;k<K;k++)
    {
      for(i1=1;i1<=mu;i1,i2=i1+1,i2+1)
      {
        M[i1,j]=M[i1,j]+C[i2,j]*var(1)^k;
      }
    }
  }
  dbprint(printlevel-voice+2,"//...e-lift of nabla(e) computed ["
    +string(timer-t)+" secs]");

  return(M,N,Vnablae);
}
///////////////////////////////////////////////////////////////////////////////

static proc mid(ideal l)
{
  int i,j,id;
  int mid=0;
  for(i=size(l);i>=1;i--)
  {
    for(j=i-1;j>=1;j--)
    {
      id=int(l[i]-l[j]);
      id=max(intvec(id,-id));
      mid=max(intvec(id,mid));
    }
  }
  return(mid);
}
///////////////////////////////////////////////////////////////////////////////

static proc decmide(matrix M,ideal eM0,list bM0)
{
  matrix M0=jet(M,0);

  dbprint(printlevel-voice+2,
    "//computing basis U of generalized eigenspaces of M0...");
  int t=timer;
  int i,j;
  matrix U,M0e;
  matrix E=freemodule(nrows(M));
  for(i=ncols(eM0);i>=1;i--)
  {
    M0e=E;
    for(j=max(bM0[i]);j>=1;j--)
    {
      M0e=M0e*(M0-eM0[i]*E);
    }
    U=syz(M0e)+U;
  }
  dbprint(printlevel-voice+2,"//...U computed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,"//transforming M to U...");
  t=timer;
  list daU=detadj(U);
  daU[2]=(1/number(daU[1]))*daU[2];
  M=daU[2]*M*U;
  dbprint(printlevel-voice+2,"//...M transformed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,
    "//computing integer differences of eigenvalues of M0...");
  t=timer;
  int k;
  intvec ideM0;
  ideM0[ncols(eM0)]=0;
  for(i=ncols(eM0);i>=1;i--)
  {
    for(j=ncols(eM0);j>=1;j--)
    {
      k=int(eM0[i]-eM0[j]);
      if(k)
      {
        if(k>0)
        {
          ideM0[i]=max(intvec(k,ideM0[i]));
        }
        else
        {
          ideM0[j]=max(intvec(-k,ideM0[j]));
        }
      }
    }
  }
  for(i,k=size(bM0),nrows(M);i>=1;i--)
  {
    for(j=sum(bM0[i]);j>=1;j--)
    {
      ideM0[k]=ideM0[i];
      k--;
    }
  }
  dbprint(printlevel-voice+2,
    "//...integer differences of eigenvalues of M0 computed ["+string(timer-t)
    +" secs]");

  dbprint(printlevel-voice+2,"//transforming M...");
  t=timer;
  for(i=nrows(M);i>=1;i--)
  {
    if(!ideM0[i])
    {
      M[i,i]=M[i,i]+1;
    }
    for(j=ncols(M);j>=1;j--)
    {
      if(ideM0[i]&&!ideM0[j])
      {
        M[i,j]=M[i,j]*var(1);
      }
      else
      {
        if(!ideM0[i]&&ideM0[j])
        {
          M[i,j]=M[i,j]/var(1);
        }
      }
    }
  }
  dbprint(printlevel-voice+2,"//...M transformed ["+string(timer-t)+" secs]");

  return(M);
}
///////////////////////////////////////////////////////////////////////////////

static proc nonqhmonodromy(poly f,int mu,int opt)
{
  pcvinit();

  dbprint(printlevel-voice+2,"//computing kappa, xi and u with "+
    "u*f^kappa=(matrix(jacob(f))*xi)[1,1]...");
  list jl=jacoblift(f);
  def kappa,xi,u=jl[1..3];
  dbprint(printlevel-voice+2,"//...kappa, xi and u computed");
  dbprint(printlevel-voice+2,"//kappa="+string(kappa));
  if(kappa==1)
  {
    dbprint(printlevel-voice+2,
      "//f quasihomogenous with respect to suitable coordinates");
  }
  else
  {
    dbprint(printlevel-voice+2,
      "//f not quasihomogenous for any choice of coordinates");
  }
  dbprint(printlevel-voice+2,"//xi=");
  dbprint(printlevel-voice+2,xi);
  dbprint(printlevel-voice+2,"//u="+string(u));

  int K,N,prevN;
  list e,P1,P2,Pe,V1,V2,Ve,Vnablae;

  dbprint(printlevel-voice+2,"//increasing K and N...");
  K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1,N,e,P1,P2,Pe,V1,V2,Ve);
  dbprint(printlevel-voice+2,"//...K and N increased");

  dbprint(printlevel-voice+2,"//computing C{f}-basis e of Brieskorn lattice "
    +"H''=Omega^(n+1)/df^dOmega^(n-1)...");
  int t=timer;
  e=pcvcv2p(quotV(V1+V2,N),0,N);
  dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,"//e=");
  dbprint(printlevel-voice+2,e);

  Pe=e;
  Ve=pcvp2cv(Pe,0,N);

  if(kappa==1)
  {
    dbprint(printlevel-voice+2,
      "//computing 0-jet M of e-matrix of t*nabla...");
    matrix M=list(MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae))[1];
    dbprint(printlevel-voice+2,"//...M computed");
  }
  else
  {
    dbprint(printlevel-voice+2,
      "//computing transformation matrix U to simple pole...");

    dbprint(printlevel-voice+2,"//computing t*nabla-stable lattice...");
    matrix M,prevU;
    matrix U=freemodule(mu)*var(1)^((mu-1)*(kappa-1));
    int i;
    dbprint(printlevel-voice+2,"//comparing with previous lattice...");
    t=timer;
    for(i=mu-1;i>=1&&size(reduce(U,std(prevU)))>0;i--)
    {
      dbprint(printlevel-voice+2,"//...compared with previous lattice ["
        +string(timer-t)+" secs]");

      dbprint(printlevel-voice+2,"//increasing K and N...");
      K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,kappa-1,N,e,P1,P2,Pe,V1,V2,Ve);
      dbprint(printlevel-voice+2,"//...K and N increased");

      dbprint(printlevel-voice+2,
        "//computing (K-1)-jet M of e-matrix of t^kappa*nabla...");
      M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae);
      dbprint(printlevel-voice+2,"//...M computed");

      prevU=U;

      dbprint(printlevel-voice+2,"//enlarging lattice...");
      t=timer;
      U=interred(jet(module(U)+module(var(1)*diff(U,var(1)))+
        module(mdivp(M*U,var(1)^(kappa-1))),(kappa-1)*(mu-1)));
      dbprint(printlevel-voice+2,"//...lattice enlarged ["+string(timer-t)
        +" secs]");

      dbprint(printlevel-voice+2,"//comparing with previous lattice...");
      t=timer;
    }
    dbprint(printlevel-voice+2,"//...compared with previous lattice ["
      +string(timer-t)+" secs]");
    dbprint(printlevel-voice+2,"//...t*nabla-stable lattice computed");

    if(ncols(U)>nrows(U))
    {
      dbprint(printlevel-voice+2,
        "//computing C{f}-basis of t*nabla-stable lattice...");
      t=timer;
      U=minbase(U);
      dbprint(printlevel-voice+2,
        "//...C{f}-basis of t*nabla-stable lattice computed ["+string(timer-t)
        +" secs]");
    }

    U=mdivp(U,var(1)^pcvmindeg(U));

    dbprint(printlevel-voice+2,"//...U computed");

    dbprint(printlevel-voice+2,
      "//computing determinant and adjoint matrix of U...");
    list daU=detadj(U);
    poly p=var(1)^min(intvec(pcvmindeg(daU[2]),pcvmindeg(daU[1])));
    daU[1]=daU[1]/p;
    daU[2]=mdivp(daU[2],p);
    dbprint(printlevel-voice+2,
      "//...determinant and adjoint matrix of U computed");

    if(K<kappa+pcvmindeg(daU[1]))
    {
      dbprint(printlevel-voice+2,"//increasing K and N...");
      K,N,P1,P2,Pe,V1,V2,Ve=
        incK(f,mu,K,kappa+pcvmindeg(daU[1])-K,N,e,P1,P2,Pe,V1,V2,Ve);
      dbprint(printlevel-voice+2,"//...K and N increased");

      dbprint(printlevel-voice+2,"//computing M...");
      M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae);
      dbprint(printlevel-voice+2,"//...M computed");
    }

    dbprint(printlevel-voice+2,"//transforming M/t^kappa to simple pole...");
    t=timer;
    M=mdivp(daU[2]*(var(1)^kappa*diff(U,var(1))+M*U),
      leadcoef(daU[1])*var(1)^(kappa+pcvmindeg(daU[1])-1));
    dbprint(printlevel-voice+2,"//...M/t^kappa transformed to simple pole ["
      +string(timer-t)+" secs]");
  }

  if(opt==0)
  {
    dbprint(printlevel-voice+2,
      "//...computing maximal integer difference delta of eigenvalues of M0");
    t=timer;
    list jd=jordan(M);
    def eM0,bM0=jd[1..2];
    int delta=mid(eM0);
    dbprint(printlevel-voice+2,"//...delta computed ["+string(timer-t)
      +" secs]");

    dbprint(printlevel-voice+2,"//delta="+string(delta));

    if(delta>0)
    {
      dbprint(printlevel-voice+2,"//increasing K and N...");
      if(kappa==1)
      {
        K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1+delta-K,N,e,P1,P2,Pe,V1,V2,Ve);
      }
      else
      {
        K,N,P1,P2,Pe,V1,V2,Ve=
          incK(f,mu,K,kappa+pcvmindeg(daU[1])+delta-K,N,e,P1,P2,Pe,V1,V2,Ve);
      }
      dbprint(printlevel-voice+2,"//...K and N increased");

      dbprint(printlevel-voice+2,"//computing M...");
      M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae);
      dbprint(printlevel-voice+2,"//...M computed");

      if(kappa>1)
      {
        dbprint(printlevel-voice+2,
          "//transforming M/t^kappa to simple pole...");
        t=timer;
        M=mdivp(invunit(daU[1]/var(1)^pcvmindeg(daU[1]),delta)*
          daU[2]*(var(1)^kappa*diff(U,var(1))+M*U),
          var(1)^(kappa+pcvmindeg(daU[1])-1));
        dbprint(printlevel-voice+2,
          "//...M/t^kappa transformed to simple pole ["+string(timer-t)
          +" secs]");
      }

      dbprint(printlevel-voice+2,"//decreasing delta...");
      M=decmide(M,eM0,bM0);
      delta--;
      dbprint(printlevel-voice+2,"//delta="+string(delta));

      while(delta>0)
      {
        jd=jordan(M);
        eM0,bM0=jd[1..2];
        M=decmide(M,eM0,bM0);
        delta--;
        dbprint(printlevel-voice+2,"//delta="+string(delta));
      }
      dbprint(printlevel-voice+2,"//...delta decreased");
    }
  }

  dbprint(printlevel-voice+2,"//computing 0-jet M0 of M...");
  matrix M0=jet(M,0);
  dbprint(printlevel-voice+2,"//...M0 computed");

  return(M0);
}
///////////////////////////////////////////////////////////////////////////////

static proc qhmonodromy(poly f,intvec w)
{
  dbprint(printlevel-voice+2,"//computing basis e of Milnor algebra...");
  int t=timer;
  ideal e=kbase(std(jacob(f)));
  dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs]");

  dbprint(printlevel-voice+2,
    "//computing Milnor number mu and quasihomogeneous degree d...");
  int mu,d=size(e),(transpose(leadexp(f))*w)[1];
  dbprint(printlevel-voice+2,"...mu and d computed");

  dbprint(printlevel-voice+2,"//computing te-matrix M of t*nabla...");
  matrix M[mu][mu];
  int i;
  for(i=mu;i>=1;i--)
  {
    M[i,i]=number((transpose(leadexp(e[i])+1)*w)[1])/d;
  }
  dbprint(printlevel-voice+2,"//...M computed");

  return(M);
}
///////////////////////////////////////////////////////////////////////////////

proc monodromy(poly f, list #)
"USAGE:   monodromy(f[,opt]); f poly, opt int
ASSUME:  The polynomial f in a series ring (local ordering) defines 
         an isolated hypersurface singularity.
RETURN:  The procedure returns a residue matrix M of the meromorphic 
         Gauss-Manin connection of the singularity defined by f 
         or an empty matrix if the assumptions are not fulfilled. 
         If opt=0 (default), exp(2*pi*i*M) is a monodromy matrix of f, 
         else, only the characteristic polynomial of exp(2*pi*i*M) coincides 
         with the characteristic polynomial of the monodromy of f.
THEORY:  The procedure uses an algorithm by Brieskorn (See E. Brieskorn, 
         manuscipta math. 2 (1970), 103-161) to compute a connection matrix of 
         the meromorphic Gauss-Manin connection up to arbitrarily high order, 
         and an algorithm of Gerard and Levelt (See R. Gerard, A.H.M. Levelt, 
         Ann. Inst. Fourier, Grenoble 23,1 (1973), pp. 157-195) to transform 
         it to a simple pole.
DISPLAY: printlevel>=1; shows comments.
EXAMPLE: example monodromy; shows an example.
"
{
  int opt;
  if(size(#)>0)
  {
    if(typeof(opt)=="int")
    {
      opt=#[1];
    }
    else
    {
      print("\\second parameter no int");
      return();
    }
   
  }

  dbprint(printlevel-voice+2,"//basering="+string(basering));

  int i;
  for(i=nvars(basering);i>=1;i--)
  {
    if(1<var(i))
    {
      i=-1;
    }
  }

  if(i<0)
  {
    print("//no series ring (local ordering)");

    matrix M[1][0];
    return(M);
  }
  else
  {
    dbprint(printlevel-voice+2,"//f="+string(f));

    dbprint(printlevel-voice+2,"//computing milnor number mu of f...");
    int t=timer;
    int mu=milnor(f);
    dbprint(printlevel-voice+2,"//...mu computed ["+string(timer-t)+" secs]");

    dbprint(printlevel-voice+2,"//mu="+string(mu));

    if(mu<=0)
    {
      if(mu==0)
      {
        print("//no singularity");
      }
      else
      {
        print("//non isolated singularity");
      }

      matrix M[1][0];
      return(M);
    }
    else
    {
      dbprint(printlevel-voice+2,"//computing weight vector w...");
      intvec w=qhweight(f);
      dbprint(printlevel-voice+2,"//...w computed");

      dbprint(printlevel-voice+2,"//w="+string(w));

      if(w==0)
      {
        dbprint(printlevel-voice+2,
          "//f not quasihomogeneous with respect to given coordinates");
        return(nonqhmonodromy(f,mu,opt));
      }
      else
      {
        dbprint(printlevel-voice+2,
          "//f quasihomogeneous with respect to given coordinates");
        return(qhmonodromy(f,w));
      }
    }
  }
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0,(x,y),ds;
  poly f=x2y2+x6+y6;
  matrix M=monodromy(f);
  print(M);
}
///////////////////////////////////////////////////////////////////////////////

proc H''basis(poly f)
"USAGE:   H''basis(f); f poly
ASSUME:  The polynomial f in a series ring (local ordering) defines 
         an isolated hypersurface singularity.
RETURN:  The procedure returns a list of representatives of a C{f}-basis of the
         Brieskorn lattice H''=Omega^(n+1)/df^dOmega^(n-1).
THEORY:  H'' is a free C{f}-module of rank milnor(f).
DISPLAY: printlevel>=1; shows comments.
EXAMPLE: example H''basis; shows an example.
"
{
  pcvinit();

  dbprint(printlevel-voice+2,"//basering="+string(basering));

  int i;
  for(i=nvars(basering);i>=1;i--)
  {
    if(1<var(i))
    {
      i=-1;
    }
  }

  if(i<0)
  {
    print("//no series ring (local ordering)");

    return(list());
  }
  else
  {
    dbprint(printlevel-voice+2,"//f="+string(f));

    dbprint(printlevel-voice+2,"//computing milnor number mu of f...");
    int t=timer;
    int mu=milnor(f);
    dbprint(printlevel-voice+2,"//...mu computed ["+string(timer-t)+" secs]");

    dbprint(printlevel-voice+2,"//mu="+string(mu));

    if(mu<=0)
    {
      if(mu==0)
      {
        print("//no singularity");
      }
      else
      {
        print("//non isolated singularity");
      }

      return(list());
    }
    else
    {
      dbprint(printlevel-voice+2,"//computing kappa, xi and u with "+
        "u*f^kappa=(matrix(jacob(f))*xi)[1,1]...");
      list jl=jacoblift(f);
      def kappa,xi,u=jl[1..3];
      dbprint(printlevel-voice+2,"//...kappa, xi and u computed");
      dbprint(printlevel-voice+2,"//kappa="+string(kappa));
      if(kappa==1)
      {
        dbprint(printlevel-voice+2,
          "//f quasihomogenous with respect to suitable coordinates");
      }
      else
      {
        dbprint(printlevel-voice+2,
          "//f not quasihomogenous for any choice of coordinates");
      }
      dbprint(printlevel-voice+2,"//xi=");
      dbprint(printlevel-voice+2,xi);
      dbprint(printlevel-voice+2,"//u="+string(u));

      int K,N,prevN;
      list e,P1,P2,Pe,V1,V2,Ve,Vnablae;

      dbprint(printlevel-voice+2,"//increasing K and N...");
      K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1,N,e,P1,P2,Pe,V1,V2,Ve);
      dbprint(printlevel-voice+2,"//...K and N increased");

      dbprint(printlevel-voice+2,
        "//computing C{f}-basis e of Brieskorn lattice "
        +"H''=Omega^(n+1)/df^dOmega^(n-1)...");
      t=timer;
      e=pcvcv2p(quotV(V1+V2,N),0,N);
      dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs]");

      dbprint(printlevel-voice+2,"//e=");
      dbprint(printlevel-voice+2,e);

      return(e);
    }
  }
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0,(x,y),ds;
  poly f=x2y2+x6+y6;
  H''basis(f);
}
///////////////////////////////////////////////////////////////////////////////