source: git/Singular/LIB/KVequiv.lib @ 33b509

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

version="$Id$";
info="
LIBRARY:  KVequiv.lib    PROCEDURES RELATED TO K_V-EQUIVALENCE
AUTHOR:   Anne Fruehbis-Krueger, anne@mathematik.uni-kl.de

OVERVIEW:
 Let (V,0) be a complete intersection singularity in (C^p,0) and
 f_0:(C^n,0) --> (C^p,0) an analytic map germ, which is viewed as a
 section ov V so that the singularity V_0=f_0^-1(V) is a pullback.
 K_V equivalence is then given by the group
     K_V={g | g(C^n x V) (subset) C^n x V} (subset) K,
 where K is the contact group of Mather. This library provides
 functionality for computing K_V tangent space, K_V normal space
 and liftable vector fields.
 A more detailed introduction to K_V equivalence can e.g. be found
 in [Damon,J.: On the legacy of free divisors, Amer.J.Math. 120,453-492]

PROCEDURES:
 derlogV(iV);                   derlog(V(iV))
 KVtangent(I,rname,dername,k)   K_V tangent space to given singularity
 KVversal(KVtan,I,rname,idname) K_V versal family
 KVvermap(KVtan,I)              section inducing K_V versal family
 lft_vf(I,rname,idname)         liftable vector fields

REMARKS:
 * monomial ordering should be of type (c,...)
 * monomial ordering should be local on the original (2) rings

SEE ALSO: sing_lib, deform_lib, spcurve_lib
";
////////////////////////////////////////////////////////////////////////////
// REQUIRED LIBRARIES
////////////////////////////////////////////////////////////////////////////

// first the ones written in Singular
LIB "poly.lib";

// then the ones written in C/C++
LIB("loctriv.so");

////////////////////////////////////////////////////////////////////////////
// PROCEDURES
////////////////////////////////////////////////////////////////////////////

proc derlogV(ideal iV)
"USAGE:  @code{derlogV(iV)};   @code{iV} ideal
RETURN:  matrix whose columns generate derlog(V(iV)),
         i.e. the module of vector fields on (C^p,0) tangent to V
EXAMPLE: @code{example derlogV}; shows an example
"
{
//--------------------------------------------------------------------------
// Compute jacobian matrix of iV and add all iV[i]*gen(j) as extra columns
//--------------------------------------------------------------------------
  int j;
  def jiV=jacob(iV);
  module mmV=jiV;
  for(int i=1;i<=size(iV);i++)
  {
    for(j=1;j<=size(iV);j++)
    {
      mmV=mmV,iV[i]*gen(j);
    }
  }
//--------------------------------------------------------------------------
// The generators of derlog(V) are given by the part of the syzygy matrix
// of mmV which deals with the jacobian matrix
//--------------------------------------------------------------------------
  def smmV=syz(mmV);
  matrix smaV=matrix(smmV);
  matrix smV[nvars(basering)][ncols(smaV)]=
                     smaV[1..nvars(basering),1..ncols(smaV)];
  return(smV);
}
example
{ "EXAMPLE:";echo=2;
  ring r=0,(a,b,c,d,e,f),ds;
  ideal i=ad-bc,af-be,cf-de;
  def dV=derlogV(i);
  print(dV);
}
////////////////////////////////////////////////////////////////////////////

proc KVtangent(ideal mapi,string rname,string dername,list #)
"USAGE:   @code{KVtangent(I,rname,dername[,k])}; @code{I} ideal
                                                 @code{rname,dername} strings
                                                 @code{[k]} int
RETURN:   K_V tangent space to a singularity given as a section of a
          model singularity
NOTE:     The model singularity lives in the ring given by rname and
          its derlog(V) is given by dername in that ring. The section is
          specified by the generators of mapi. If k is given, the first k
          variables are used as variables, the remaining ones as parameters
EXAMPLE:  @code{example KVtangent}; shows an example
"
{
//--------------------------------------------------------------------------
// Sanity Checks
//--------------------------------------------------------------------------
  if(size(#)==0)
  {
    int k=nvars(basering);
  }
  else
  {
    if(typeof(#[1])=="int")
    {
      int k=#[1];
    }
    else
    {
      int k=nvars(basering);
    }
  }
  def baser=basering;
  string teststr="setring " + rname + ";";
  execute(teststr);
  if(nameof(basering)!=rname)
  {
    ERROR("rname not name of a ring");
  }
  teststr="string typeder=typeof(" + dername + ");";
  execute(teststr);
  if((typeder!="matrix")&&(typeder!="module"))
  {
    ERROR("dername not name of a matrix or module in rname");
  }
  setring(baser);
  if((k > nvars(basering))||(k < 1))
  {
    ERROR("k should be between 1 and the number of variables");
  }
//--------------------------------------------------------------------------
// Define the map giving the section and use it for substituting the
// variables of the ring rname by the entries of mapi in the matrix
// given by dername
//--------------------------------------------------------------------------
  setring baser;
  string mapstr="map f0=" + rname + ",";
  for(int i=1;i<ncols(mapi);i++)
  {
    mapstr=mapstr + string(mapi[i]) + ",";
  }
  mapstr=mapstr + string(mapi[ncols(mapi)]) + ";";
  execute(mapstr);
  string derstr="def derim=f0(" + dername + ");";
  execute(derstr);
//---------------------------------------------------------------------------
// Form the derivatives of mapi by the first k variables
//---------------------------------------------------------------------------
  matrix jmapi[ncols(mapi)][k];
  for(i=1;i<=k;i++)
  {
    jmapi[1..nrows(jmapi),i]=diff(mapi,var(i));
  }
//---------------------------------------------------------------------------
// Put everything together to get the tangent space
//---------------------------------------------------------------------------
  string nvstr="int nvmodel=nvars(" + rname + ");";
  execute(nvstr);
  matrix M[nrows(derim)][ncols(derim)+k];
  M[1..nrows(M),1..ncols(derim)]=derim[1..nrows(derim),1..ncols(derim)];
  M[1..nrows(M),(ncols(derim)+1)..ncols(M)]=
            jmapi[1..nrows(M),1..k];
  return(M);
}
example
{ "EXAMPLE:";echo=2;
  ring ry=0,(a,b,c,d),ds;
  ideal idy=ab,cd;
  def dV=derlogV(idy);
  echo=1; export ry; export dV; echo=2;
  ring rx=0,(x,y,z),ds;
  ideal mi=x-z+2y,x+y-z,y-x-z,x+2z-3y;
  def M=KVtangent(mi,"ry","dV");
  print(M);
  M[1,5];
  echo=1; kill ry;
}
/////////////////////////////////////////////////////////////////////////////

proc KVversal(matrix KVtan, ideal mapi, string rname, string idname)
"USAGE:   @code{KVversal(KVtan,I,rname,idname)};  @code{KVtan} matrix
                                                  @code{I} ideal
                                                  @code{rname,idname} strings
RETURN:   list; The first entry of the list is the new ring in which the
          K_V versal family lives, the second is the name of the ideal
          describing a K_V versal family of a singularity given as section
          of a model singularity (which was specified as idname in rname)
NOTE:     The section is given by the generators of I, KVtan is the matrix
          describing the K_V tangent space to the singularity (as returned
          by KVtangent). rname denotes the ring in which the model
          singularity lives, and idname is the name of the ideal in this ring
          defining the singularity.
EXAMPLE:  @code{example KVversal}; shows an example
"
{
//---------------------------------------------------------------------------
// Sanity checks
//---------------------------------------------------------------------------
  def baser=basering;
  string teststr="setring " + rname + ";";
  execute(teststr);
  if(nameof(basering)!=rname)
  {
    ERROR("rname not name of a ring");
  }
  teststr="string typeid=typeof(" + idname + ");";
  execute(teststr);
  if(typeid!="ideal")
  {
    ERROR("idname not name of an ideal in rname");
  }
  setring baser;
//---------------------------------------------------------------------------
// Find a monomial basis of the K_V normal space
// and check whether we can define new variables A(i)
//---------------------------------------------------------------------------
  module KVt=KVtan;
  module KVts=std(KVt);
  module kbKVt=kbase(KVts);
  for(int i=1; i<=size(kbKVt); i++)
  {
    if(rvar(A(i)))
    {
      int jj=-1;
      break;
    }
  }
  if (defined(jj)>1)
  {
    if (jj==-1)
    {
      ERROR("Your ring contains a variable A(i)!");
    }
  }
//---------------------------------------------------------------------------
// Extend our current ring by adjoining the correct number of variables
// A(i) for the parameters and copy our objects to this ring
//---------------------------------------------------------------------------
  def rbas=basering;
  ring rtemp=0,(A(1..size(kbKVt))),(c,dp);
  def rpert=rbas + rtemp;
  setring rpert;
  def mapi=imap(rbas,mapi);
  def kbKVt=imap(rbas,kbKVt);
  matrix mapv[ncols(mapi)][1]=mapi;   // I hate the conversion from ideal
  vector mapV=mapv[1];                // to vector
//---------------------------------------------------------------------------
// Build up the map of the perturbed section and apply it to the ideal
// idname
//---------------------------------------------------------------------------
  for(i=1;i<=size(kbKVt);i++)
  {
    mapV=mapV+A(i)*kbKVt[i];
  }

  string mapstr="map fpert=" + rname + ",";
  for(int i=1;i<size(mapV);i++)
  {
    mapstr=mapstr + string(mapV[i]) + ",";
  }
  mapstr=mapstr + string(mapV[size(mapV)]) + ";";
  execute(mapstr);
  string idstr="ideal Ipert=fpert(" + idname + ");";
  execute(idstr);
//---------------------------------------------------------------------------
// Return our new ring and the name of the perturbed ideal
//---------------------------------------------------------------------------
  export Ipert;
  list retlist=rpert,"Ipert";
  return(retlist);
}
example
{ "EXAMPLE:";echo=2;
  ring ry=0,(a,b,c,d),ds;
  ideal idy=ab,cd;
  def dV=derlogV(idy);
  echo=1;
  export ry; export dV; export idy; echo=2;
  ring rx=0,(x,y,z),ds;
  ideal mi=x-z+2y,x+y-z,y-x-z,x+2z-3y;
  def M=KVtangent(mi,"ry","dV");
  list li=KVversal(M,mi,"ry","idy");
  def rnew=li[1];
  setring rnew;
  `li[2]`;
  echo=1;
  setring ry; kill idy; kill dV; setring rx; kill ry;
}
/////////////////////////////////////////////////////////////////////////////

proc KVvermap(matrix KVtan, ideal mapi)
"USAGE:   @code{KVvermap(KVtan,I)};  @code{KVtan} matrix, @code{I} ideal
RETURN:   list; The first entry of the list is the new ring in which the
          versal object lives, the second specifies a map describing the
          section which yields a K_V versal family of the original
          singularity which was given as section of a model singularity
NOTE:     The section is given by the generators of I, KVtan is the matrix
          describing the K_V tangent space to the singularity (as returned
          by KVtangent).
EXAMPLE:  @code{example KVvermap}; shows an example
"
{
//---------------------------------------------------------------------------
// Find a monomial basis of the K_V normal space
// and check whether we can define new variables A(i)
//---------------------------------------------------------------------------
  module KVt=KVtan;
  module KVts=std(KVt);
  module kbKVt=kbase(KVts);
  for(int i=1; i<=size(kbKVt); i++)
  {
    if(rvar(A(i)))
    {
      int jj=-1;
      break;
    }
  }
  if (defined(jj)>1)
  {
    if (jj==-1)
    {
      ERROR("Your ring contains a variable A(i)!");
    }
  }
//---------------------------------------------------------------------------
// Extend our current ring by adjoining the correct number of variables
// A(i) for the parameters and copy our objects to this ring
//---------------------------------------------------------------------------
  def rbas=basering;
  ring rtemp=0,(A(1..size(kbKVt))),(c,dp);
  def rpert=rbas + rtemp;
  setring rpert;
  def mapi=imap(rbas,mapi);
  def kbKVt=imap(rbas,kbKVt);
  matrix mapv[ncols(mapi)][1]=mapi;
  vector mapV=mapv[1];
//---------------------------------------------------------------------------
// Build up the map of the perturbed section
//---------------------------------------------------------------------------
  for(i=1;i<=size(kbKVt);i++)
  {
    mapV=mapV+A(i)*kbKVt[i];
  }
  ideal mappert=mapV[1..size(mapV)];
//---------------------------------------------------------------------------
// Return the new ring and the name of an ideal describing the perturbed map
//---------------------------------------------------------------------------
  export mappert;
  list retlist=basering,"mappert";
  return(retlist);
}
example
{ "EXAMPLE:";echo=2;
  ring ry=0,(a,b,c,d),ds;
  ideal idy=ab,cd;
  def dV=derlogV(idy);
  echo=1;
  export ry; export dV; export idy; echo=2;
  ring rx=0,(x,y,z),ds;
  ideal mi=x-z+2y,x+y-z,y-x-z,x+2z-3y;
  def M=KVtangent(mi,"ry","dV");
  list li=KVvermap(M,mi);
  def rnew=li[1];
  setring rnew;
  `li[2]`;
  echo=1;
  setring ry; kill idy; kill dV; setring rx; kill ry;
}
/////////////////////////////////////////////////////////////////////////////

proc lft_vf(ideal mapi, string rname, string idname, intvec wv, int b, list #)
"USAGE: @code{lft_vf(I,rname,iname,wv,b[,any])}
                                       @code{I} ideal
                                       @code{wv} intvec
                                       @code{b} int
                                       @code{rname,iname} strings
                                       @code{[any]} def
RETURN: list
        [1]: ring in which objects specified by the strings [2] and [3] live
        [2]: name of ideal describing the liftable vector fields -
             computed up to order b in the parameters
        [3]: name of basis of the K_V-normal space of the original singularity
        [4]: (if 6th argument is given)
             ring in which the reduction of the liftable vector fields has
             taken place.
        [5]: name of liftable vector fields in ring [4]
        [6]: name of ideal we are using for reduction of [5] in [4]
ASSUME: input is assumed to be quasihomogeneous in the following sense:
        there are weights for the variables in the current basering
        such that, after plugging in mapi[i] for the i-th variable of the
        ring rname in the ideal idname, the resulting expression is
        quasihomogeneous; wv specifies the weight vector of the ring rname.
        b is the degree bound up in the perturbation parameters up to which
        computations are performed.
NOTE:   the original ring should not contain any variables of name
        A(i) or e(j)
EXAMPLE:@code{example lft_vf;} gives an example
"
{
//---------------------------------------------------------------------------
// Sanity checks
//---------------------------------------------------------------------------
  def baser=basering;
  def qid=maxideal(1);
  string teststr="setring " + rname + ";";
  execute(teststr);
  if(nameof(basering)!=rname)
  {
    ERROR("rname not name of a ring");
  }
  def ry=basering;
  teststr="string typeid=typeof(" + idname + ");";
  execute(teststr);
  if(typeid!="ideal")
  {
    ERROR("idname not name of an ideal in rname");
  }
  setring baser;
  for(int i=1; i<=ncols(mapi); i++)
  {
    if(rvar(e(i)))
    {
      int jj=-1;
      break;
    }
  }
  if (defined(jj)>1)
  {
    if (jj==-1)
    {
      ERROR("Your ring contains a variable e(j)!");
    }
  }
  setring ry;
//---------------------------------------------------------------------------
// first prepare derlog(V) for the model singularity
// and set the correct weights
//---------------------------------------------------------------------------
  def @dV=derlogV(`idname`);
  export(@dV);
  setring baser;
  map maptemp=`rname`,mapi;
  def tempid=maptemp(`idname`);
  intvec ivm=qhweight(tempid);
  string ringstr="ring baserw=" + charstr(baser) + ",(" + varstr(baser) +
                 "),(c,ws(" + string(ivm) + "));";
  execute(ringstr);
  def mapi=imap(baser,mapi);
//---------------------------------------------------------------------------
// compute the unperturbed K_V tangent space
// and check K_V codimension
//---------------------------------------------------------------------------
  def KVt=KVtangent(mapi,rname,"@dV",nvars(basering));
  def sKVt=std(KVt);
  if(dim(sKVt)>0)
  {
    ERROR("K_V-codimension not finite");
  }
//---------------------------------------------------------------------------
// Construction of the versal family
//---------------------------------------------------------------------------
  list lilit=KVvermap(KVt,mapi);
  def rpert=lilit[1];
  setring rpert;
  def mapipert=`lilit[2]`;
  def KVt=imap(baserw,KVt);
  def mapi=imap(baserw,mapi);
  def KVtpert=KVtangent(mapipert,rname,"@dV",nvars(baser));
//---------------------------------------------------------------------------
// put the unperturbed and the perturbed tangent space into a module
// (1st component unperturbed) and run a groebner basis computation
// which only considers spolys with non-vanishing first component
//---------------------------------------------------------------------------
  def rxa=basering;
  string rchange="ring rexa=" + charstr(basering) + ",(e(1.." +
                 string(ncols(mapi)) + ")," + varstr(basering) +
                 "),(c,ws(" + string((-1)*wv) + "," + string(ivm) + "),dp);";
  execute(rchange);
  def mapi=imap(rxa,mapi);
  ideal eid=e(1..ncols(mapi));            // for later use
  def KVt=imap(rxa,KVt);
  def KVtpert=imap(rxa,KVtpert);
  intvec iv=1..ncols(mapi);
  ideal KVti=mod2id(KVt,iv);
//----------------------------------------------------------------------------
// small intermezzo (here because we did not have all input any earlier)
// get kbase of KVti for later use and determine an
// integer l such that m_x^l*(e_1,\dots,e_r) lies in KVt
//----------------------------------------------------------------------------
  ideal sKVti=std(KVti);
  ideal lsKVti=lead(sKVti);
  module tmpmo=id2mod(lsKVti,iv);
  setring baser;
  def tmpmo=imap(rexa,tmpmo);
  attrib(tmpmo,"isSB",1);
  module kbKVt=kbase(tmpmo);
  setring rexa;
  def kbKVt=imap(baser,kbKVt);
  ideal kbKVti=mod2id(kbKVt,iv);
  def qid=imap(baser,qid);
  intvec qiv;
  for(i=1;i<=ncols(qid);i++)
  {
    qiv[rvar(qid[i])]=1;
  }
  int counter=1;
  while(size(reduce(lsKVti,std(jet(lsKVti,i,qiv))))!=0)
  {
    counter++;
  }
//----------------------------------------------------------------------------
// end of intermezzo
// proceed to the previously announced Groebner basis computation
//----------------------------------------------------------------------------
  ideal KVtpi=mod2id(KVtpert,iv);
  export(KVtpi);
  matrix Eing[2][ncols(KVti)]=KVti,KVtpi;
  module EinMo=Eing;
  EinMo=EinMo,eid^2*gen(1),eid^2*gen(2);
  module Ausg=Loctriv::kstd(EinMo,1);
//---------------------------------------------------------------------------
// * collect those elements of Ausg for which the first component is non-zero
//   into mx and the others into mt
// * cut off the first component
// * find appropriate weights for the reduction
//---------------------------------------------------------------------------
  intvec eiv;
  for(i=1;i<=ncols(eid);i++)
  {
    eiv[rvar(eid[i])]=1;
  }
  if(size(reduce(var(nvars(basering)),std(eid)))!=0)
  {
    eiv[nvars(basering)]=0;
  }
  module Aus2=jet(Ausg,1,eiv);
  Aus2=simplify(Aus2,2);
  ideal mx;
  ideal mt;
  int ordmax,ordmin;
  int ordtemp;
  for (i=1;i<=size(Aus2);i++)
  {
    if(Aus2[1,i]!=0)
    {
      mx=mx,Aus2[2,i];
      ordtemp=ord(lead(Aus2[1,i]));
      if(ordtemp>ordmax)
      {
        ordmax=ordtemp;
      }
      else
      {
        if(ordtemp<ordmin)
        {
          ordmin=ordtemp;
        }
      }
    }
    else
    {
      mt=mt,Aus2[2,i];
    }
  }
//---------------------------------------------------------------------------
// * change weights of the A(i) such that Aus2[1,i] and Aus2[2,i] have the
//   same leading term, if the first one is non-zero
// * reduce mt by mx
// * find l such that (x_1,...,x_n)^l * eid can be used instead of noether
//   which we have to avoid because we are playing with the weights
//---------------------------------------------------------------------------
  intvec oiv;
  for(i=1;i<=(nvars(basering)-nvars(baser)-size(eid));i++)
  {
    oiv[i]=2*(abs(ordmax)+abs(ordmin));
  }
  mx=jet(mx,counter*(b+1),qiv);
  rchange="ring rexaw=" + charstr(basering) + ",(" + varstr(basering) +
                      "),(c,ws(" + string((-1)*wv) + "," + string(ivm) +
                      "," + string(oiv) + "));";
  execute(rchange);
  ideal qid=imap(rexa,qid);
  def eid=imap(rexa,eid);
  def mx=imap(rexa,mx);
  attrib(mx,"isSB",1);
  def mto=imap(rexa,mt);
  ideal Aid=A(1..size(oiv));
  intvec Aiv;
  for(i=1;i<=ncols(Aid);i++)
  {
    Aiv[rvar(Aid[i])]=1;
  }
  intvec riv=(b+1)*qiv+(b+2)*counter*Aiv;
  def mt=mto;
  for(i=1;i<=counter+1;i++)
  {
    mt=mt,mto*qid^i;
  }
  mt=jet(mt,(b+1)*(b+2)*counter,riv);
  mt=jet(mt,1,eiv);
  mt=simplify(mt,10);
  module mmx=module(mx);
  attrib(mmx,"isSB",1);
  for(i=1;i<=ncols(mt);i++)
  {
    if(defined(watchProgress))
    {
      "reducing mt[i], i="+string(i);
    }
   mt[i]=system("locNF",vector(mt[i]),mmx,
                        (b+1)*(b+2)*counter,riv)[1][1,1];
  }
  mt=simplify(mt,10);
//----------------------------------------------------------------------------
// return the results by returning the ring and the names of the desired
// modules in the ring
// (if the list # is not empty, then we want to return this ring as well)
//----------------------------------------------------------------------------
  if(size(#)!=0)
  {
    export mt;
    export mx;
  }
  setring rexa;
  def mtout=imap(rexaw,mt);
  kbKVti=jet(kbKVti,1,eiv);
  kbKVti=simplify(kbKVti,2);
  intvec rediv;
  int j=1;
  for(i=1;i<=size(qiv);i++)
  {
    if(qiv[i]!=0)
    {
      rediv[j]=i;
      j++;
    }
  }
  list templi=subrInterred(kbKVti,mtout,rediv);
  mtout=jet(templi[3],b+1,Aiv);
  export mtout;
  export kbKVti;
  list result;
  result[1]=rexa;
  result[2]="mtout";
  result[3]="kbKVti";
  if(size(#)!=0)
  {
    result[4]=rexaw;
    result[5]="mt";
    result[6]="mx";
  }
  export rexa;
  keepring rexa;
  return(result);
}
example
{ "EXAMPLE:";echo=2;
  ring ry=0,(a,b,c,d),ds;
  ideal idy=ab,cd;
  def dV=derlogV(idy);
  echo=1;
  export ry; export dV; export idy; echo=2;
  ring rx=0,(x,y,z),ds;
  ideal mi=x-z+2y,x+y-z,y-x-z,x+2z-3y;
  intvec wv=1,1,1,1;
  def M=KVtangent(mi,"ry","dV");
  list li=lft_vf(mi,"ry","idy",wv,5);
  def rr=li[1];
  setring rr;
  `li[2]`;
  `li[3]`;
  echo=1;
  setring ry; kill idy; kill dV; setring rx; kill ry;
}
//////////////////////////////////////////////////////////////////////////////
// STATIC PROCEDURES
//////////////////////////////////////////////////////////////////////////////
static
proc abs(int c)
"absolute value
"
{
  if(c>=0){ return(c);}
  else{ return(-c);}
}
////////////////////////////////////////////////////////////////////////////