source: git/Singular/LIB/sing.lib

////////////////////////////////////////////////////////////////////////////
version="version sing.lib 4.4.0.0 Nov_2023 "; // $Id$
category="Singularities";
info="
LIBRARY:  sing.lib      Invariants of Singularities
AUTHORS:  Gert-Martin Greuel, email: greuel@mathematik.uni-kl.de @*
          Bernd Martin, email: martin@math.tu-cottbus.de

PROCEDURES:
 codim(id1, id2);       vector space dimension of id2/id1 if finite
 deform(i);             infinitesimal deformations of ideal i
 dim_slocus(i);         dimension of singular locus of ideal i
 is_active(f,id);       is polynomial f an active element mod id? (id ideal/module)
 is_ci(i);              is ideal i a complete intersection?
 is_is(i);              is ideal i an isolated singularity?
 is_reg(f,id);          is polynomial f a regular element mod id? (id ideal/module)
 is_regs(i[,id]);       are gen's of ideal i regular sequence modulo id?
 locstd(i);             SB for local degree ordering without cancelling units
 milnor(i);             milnor number of ideal i; (assume i is ICIS in nf)
 nf_icis(i);            generic combinations of generators; get ICIS in nf
 slocus(i);             ideal of singular locus of ideal i
 qhspectrum(f,w);       spectrum numbers of w-homogeneous polynomial f
 Tjurina(i);            SB of Tjurina module of ideal i (assume i is ICIS)
 tjurina(i);            Tjurina number of ideal i (assume i is ICIS)
 T_1(i);                T^1-module of ideal i
 T_2((i);               T^2-module of ideal i
 T_12(i);               T^1- and T^2-module of ideal i
 tangentcone(id);       compute tangent cone of id

";

LIB "inout.lib";
LIB "random.lib";
LIB "primdec.lib";
///////////////////////////////////////////////////////////////////////////////

proc deform (ideal id)
"USAGE:   deform(id); id=ideal or poly
RETURN:  matrix, columns are kbase of infinitesimal deformations
EXAMPLE: example deform; shows an example
"
{
   list L=T_1(id,"");
   def K=L[1]; attrib(K,"isSB",1);
   return(matrix(L[2])*matrix(kbase(K)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r   = 32003,(x,y,z),ds;
   ideal i  = xy,xz,yz;
   matrix T = deform(i);
   print(T);
   print(deform(x3+y5+z2));
}
///////////////////////////////////////////////////////////////////////////////

proc dim_slocus (ideal i)
"USAGE:   dim_slocus(i);  i ideal or poly
RETURN:  dimension of singular locus of i
EXAMPLE: example dim_slocus; shows an example
"
{
   return(dim(std(slocus(i))));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 32003,(x,y,z),ds;
   ideal i = x5+y6+z6,x2+2y2+3z2;
   dim_slocus(i);
}
///////////////////////////////////////////////////////////////////////////////

proc is_active (poly f,def id)
"USAGE:   is_active(f,id); f poly, id ideal or module
RETURN:  1 if f is an active element modulo id (i.e. dim(id)=dim(id+f*R^n)+1,
         if id is a submodule of R^n) resp. 0 if f is not active.
         The basering may be a quotient ring
NOTE:    regular parameters are active but not vice versa (id may have embedded
         components). proc is_reg tests whether f is a regular parameter
EXAMPLE: example is_active; shows an example
"
{
   if( size(id)==0 ) { return(1); }
   if( typeof(id)=="ideal" ) { ideal m=f; }
   if( typeof(id)=="module" ) { module m=f*freemodule(nrows(id)); }
   return(dim(std(id))-dim(std(id+m)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r   =32003,(x,y,z),ds;
   ideal i  = yx3+y,yz3+y3z;
   poly f   = x;
   is_active(f,i);
   qring q  = std(x4y5);
   poly f   = x;
   module m = [yx3+x,yx3+y3x];
   is_active(f,m);
}
///////////////////////////////////////////////////////////////////////////////

proc is_ci (ideal i)
"USAGE:   is_ci(i); i ideal
RETURN:  intvec = sequence of dimensions of ideals (j[1],...,j[k]), for
         k=1,...,size(j), where j is minimal base of i. i is a complete
         intersection if last number equals nvars-size(i)
NOTE:    dim(0-ideal) = -1. You may first apply simplify(i,10); in order to
         delete zeroes and multiples from set of generators
         printlevel >=0: display comments (default)
EXAMPLE: example is_ci; shows an example
"
{
   int n; intvec dimvec; ideal id;
   i=minbase(i);
   int s = ncols(i);
   int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
//--------------------------- compute dimensions ------------------------------
   for( n=1; n<=s; n=n+1 )
   {
      id = i[1..n];
      dimvec[n] = dim(std(id));
   }
   n = dimvec[s];
//--------------------------- output ------------------------------------------
   if( n+s != nvars(basering) )
   { dbprint(p,"// no complete intersection"); }
   if( n+s == nvars(basering) )
   { dbprint(p,"// complete intersection of dim "+string(n)); }
   dbprint(p,"// dim-sequence:");
   return(dimvec);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;                // display comments
   ring r     = 32003,(x,y,z),ds;
   ideal i    = x4+y5+z6,xyz,yx2+xz2+zy7;
   is_ci(i);
   i          = xy,yz;
   is_ci(i);
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc is_is (ideal i)
"USAGE:   is_is(id);  id ideal or poly
RETURN:  intvec = sequence of dimensions of singular loci of ideals
         generated by id[1]..id[i], k = 1..size(id); @*
         dim(0-ideal) = -1;
         id defines an isolated singularity if last number is 0
NOTE:    printlevel >=0: display comments (default)
EXAMPLE: example is_is; shows an example
"
{
  int l; intvec dims; ideal j;
  int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
//--------------------------- compute dimensions ------------------------------
   for( l=1; l<=ncols(i); l=l+1 )
   {
     j = i[1..l];
     dims[l] = dim(std(slocus(j)));
   }
   dbprint(p,"// dim of singular locus = "+string(dims[size(dims)]),
             "// isolated singularity if last number is 0 in dim-sequence:");
   return(dims);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r     = 32003,(x,y,z),ds;
   ideal i    = x2y,x4+y5+z6,yx2+xz2+zy7;
   is_is(i);
   poly f     = xy+yz;
   is_is(f);
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc is_reg (poly f,def id)
"USAGE:   is_reg(f,id); f poly, id ideal or module
RETURN:  1 if multiplication with f is injective modulo id, 0 otherwise
NOTE:    Let R be the basering and id a submodule of R^n. The procedure checks
         injectivity of multiplication with f on R^n/id. The basering may be a
         quotient ring.
EXAMPLE: example is_reg; shows an example
"
{
   if( f==0 ) { return(0); }
   int d,ii;
   def q = quotient(id,ideal(f));
   id=std(id);
   d=size(q);
   for( ii=1; ii<=d; ii=ii+1 )
   {
      if( reduce(q[ii],id)!=0 )
      { return(0); }
   }
   return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 32003,(x,y),ds;
   ideal i = x8,y8;
   ideal j = (x+y)^4;
   i       = intersect(i,j);
   poly f  = xy;
   is_reg(f,i);
}
///////////////////////////////////////////////////////////////////////////////

proc is_regs (ideal i, list #)
"USAGE:   is_regs(i[,id]); i poly, id ideal or module (default: id=0)
RETURN:  1 if generators of i are a regular sequence modulo id, 0 otherwise
NOTE:    Let R be the basering and id a submodule of R^n. The procedure checks
         injectivity of multiplication with i[k] on R^n/id+i[1..k-1].
         The basering may be a quotient ring.
         printlevel >=0: display comments (default)
         printlevel >=1: display comments during computation
EXAMPLE: example is_regs; shows an example
"
{
   int d,ii,r;
   int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
   if( size(#)==0 ) { ideal id; }
   else { def id=#[1]; }
   if( size(i)==0 ) { return(0); }
   d=size(i);
   if( typeof(id)=="ideal" ) { ideal m=1; }
   if( typeof(id)=="module" ) { module m=freemodule(nrows(id)); }
   for( ii=1; ii<=d; ii=ii+1 )
   {
      if( p>=2 )
      { "// checking whether element",ii,"is regular mod 1 ..",ii-1; }
      if( is_reg(i[ii],id)==0 )
      {
        dbprint(p,"// elements 1.."+string(ii-1)+" are regular, " +
                string(ii)+" is not regular mod 1.."+string(ii-1));
         return(0);
      }
      id=id+i[ii]*m;
   }
   if( p>=1 ) { "// elements are a regular sequence of length",d; }
   return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r1    = 32003,(x,y,z),ds;
   ideal i    = x8,y8,(x+y)^4;
   is_regs(i);
   module m   = [x,0,y];
   i          = x8,(x+z)^4;;
   is_regs(i,m);
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc milnor (ideal i)
"USAGE:   milnor(i); i ideal or poly
RETURN:  Milnor number of i, if i is ICIS (isolated complete intersection
         singularity) in generic form, resp. -1 if not
NOTE:    use proc nf_icis to put generators in generic form
         printlevel >=1: display comments
EXAMPLE: example milnor; shows an example
"
{
  i = simplify(i,10);     //delete zeroes and multiples from set of generators
  int n = size(i);
  int l,q,m_nr;  ideal t;  intvec disc;
  int p = printlevel-voice+2;             // p=printlevel+1 (default: p=0)
//---------------------------- hypersurface case ------------------------------
  if( n==1 or n==0 )
  {
     i = std(jacob(i[1]));
     m_nr = vdim(i);
     if( m_nr<0 and p>=1 ) { "// Milnor number is infinite"; }
     return(m_nr);
  }
//------------ isolated complete intersection singularity (ICIS) --------------
  for( l=n; l>0; l=l-1)
  {   t      = minor(jacob(i),l);
      i[l]   = 0;
      q      = vdim(std(i+t));
      disc[l]= q;
      if( q ==-1 )
      {  if( p>=1 )
            {  "// not in generic form or no ICIS; use proc nf_icis to put";
            "// generators in generic form and then try milnor again!";  }
         return(q);
      }
      m_nr = q-m_nr;
  }
//---------------------------- change sign ------------------------------------
  if (m_nr < 0) { m_nr=-m_nr; }
  if( p>=1 ) { "//sequence of discriminant numbers:",disc; }
  return(m_nr);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 2;
   ring r     = 32003,(x,y,z),ds;
   ideal j    = x5+y6+z6,x2+2y2+3z2,xyz+yx;
   milnor(j);
   poly f     = x7+y7+(x-y)^2*x2y2+z2;
   milnor(f);
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc nf_icis (ideal i)
"USAGE:   nf_icis(i); i ideal
RETURN:  ideal = generic linear combination of generators of i if i is an ICIS
         (isolated complete intersection singularity), return i if not
NOTE:    this proc is useful in connection with proc milnor
         printlevel >=0: display comments (default)
EXAMPLE: example nf_icis; shows an example
"
{
   i = simplify(i,10);  //delete zeroes and multiples from set of generators
   int p,b = 100,0;
   int n = size(i);
   matrix mat=freemodule(n);
   int P = printlevel-voice+3;  // P=printlevel+1 (default: P=1)
//---------------------------- test: complete intersection? -------------------
   intvec sl = is_ci(i);
   if( n+sl[n] != nvars(basering) )
   {
      dbprint(P,"// no complete intersection");
      return(i);
   }
//--------------- test: isolated singularity in generic form? -----------------
   sl = is_is(i);
   if ( sl[n] != 0 )
   {
      dbprint(P,"// no isolated singularity");
      return(i);
   }
//------------ produce generic linear combinations of generators --------------
   int prob;
   while ( sum(sl) != 0 )
   {  prob=prob+1;
      p=p-25; b=b+10;
      i = genericid(i,p,b);          // proc genericid from random.lib
      sl = is_is(i);
   }
   dbprint(P,"// ICIS in generic form after "+string(prob)+" genericity loop(s)");
   return(i);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r     = 32003,(x,y,z),ds;
   ideal i    = x3+y4,z4+yx;
   nf_icis(i);
   ideal j    = x3+y4,xy,yz;
   nf_icis(j);
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc slocus(ideal i)
"USAGE:   slocus(i);  i ideal
RETURN:  ideal of singular locus of i. Quotient rings and rings with integer coefficients are currently not supported.
EXAMPLE: example slocus; shows an example
"
{
  // quotient rings currently not supported
  ASSUME( 0, 0==isQuotientRing(basering) );
  // integer coefficient rings currently not supported
  ASSUME( 0, hasFieldCoefficient(basering) );


  def R=basering;
  int j,k;
  ideal res;

  if(ord_test(basering)!=1)
  {
     string va=varstr(basering);
     if( size( parstr(basering))>0){va=va+","+parstr(basering);}
     execute ("ring S = ("+charstr(basering)+"),("+va+"),dp;");
     ideal i=imap(R,i);
     list l=equidim(i);
     setring R;
     list l=imap(S,l);
  }
  else
  {
     list l=equidim(i);
  }
  int n=size(l);
  if (n==1){return(slocusEqi(i));}
  res=slocusEqi(l[1]);
  for(j=2;j<=n;j++){res=intersect(res,slocusEqi(l[j]));}
  for(j=1;j<n;j++)
  {
     for(k=j+1;k<=n;k++){res=intersect(res,l[j]+l[k]);}
  }
  return(res);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 0,(u,v,w,x,y,z),dp;
   ideal i = wx,wy,wz,vx,vy,vz,ux,uy,uz,y3-x2;;
   slocus(i);
}
///////////////////////////////////////////////////////////////////////////////

static proc slocusEqi (ideal i)
"USAGE:   slocus(i);  i ideal
RETURN:  ideal of singular locus of i if i is pure dimensional
NOTE:    this proc returns i and c-minors of jacobian ideal of i where c is the
         codimension of i. Hence, if i is not pure dimensional, slocus may
         return an ideal such that its 0-locus is strictly contained in the
         singular locus of i
EXAMPLE: example slocus; shows an example
"
{
  ideal ist=std(i);
  if ( size(ist)==0 ) // we have a zero ideal
  {
     // the zero locus of the zero ideal is nonsingular
     return( ideal(1) ) ;
  }
  if( deg( ist[1] ) == 0 ) // the ideal has a constant generator
  {
    return(ist);
  }
  int cod  = nvars(basering) - dim(ist);
  i        = i + minor( jacob(i), cod );
  return(i);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 0,(x,y,z),ds;
   ideal i = x5+y6+z6,x2+2y2+3z2;
   slocus(i);
}
///////////////////////////////////////////////////////////////////////////////

proc qhspectrum (poly f, intvec w)
"USAGE:   qhspectrum(f,w);  f=poly, w=intvec
ASSUME:  f is a weighted homogeneous isolated singularity w.r.t. the weights
         given by w; w must consist of as many positive integers as there
         are variables of the basering
COMPUTE: the spectral numbers of the w-homogeneous polynomial f, computed in a
         ring of characteristic 0
RETURN:  intvec  d,s1,...,su  where:
         d = w-degree(f)  and  si/d = i-th spectral-number(f)
         No return value if basering has parameters or if f is no isolated
         singularity, displays a warning in this case.
EXAMPLE: example qhspectrum; shows an example
"
{
   int i,d,W;
   intvec sp;
   def r   = basering;
   if( find(charstr(r),",")!=0 )
   {
       "// coefficient field must not have parameters!";
       return();
    }
   ring s  = 0,x(1..nvars(r)),ws(w);
   map phi = r,maxideal(1);
   poly f  = phi(f);
   d       = ord(f);
   W       = sum(w)-d;
   ideal k = std(jacob(f));
   if( vdim(k) == -1 )
   {
       "// f is no isolated singularity!";
       return();
    }
   k = kbase(k);
   for (i=1; i<=size(k); i++)
   {
      sp[i]=W+ord(k[i]);
   }
   list L  = sort(sp);
   sp      = d,L[1];
   return(sp);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r;
   poly f=x3+y5+z2;
   intvec w=10,6,15;
   qhspectrum(f,w);
   // the spectrum numbers are:
   // 1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30
}
///////////////////////////////////////////////////////////////////////////////

proc Tjurina (def id, list #)
"USAGE:   Tjurina(id[,<any>]);  id=ideal or poly
ASSUME:  id=ICIS (isolated complete intersection singularity)
RETURN:  standard basis of Tjurina-module of id,
         of type module if id=ideal, resp. of type ideal if id=poly.
         If a second argument is present (of any type) return a list: @*
           [1] = Tjurina number,
           [2] = k-basis of miniversal deformation,
           [3] = SB of Tjurina module,
           [4] = Tjurina module
DISPLAY: Tjurina number if printlevel >= 0 (default)
NOTE:    Tjurina number = -1 implies that id is not an ICIS
EXAMPLE: example Tjurina; shows examples
"
{
//---------------------------- initialisation ---------------------------------
  def i = simplify(id,10);
  int tau,n = 0,size(i);
  if( size(ideal(i))==1 ) { def m=i; }  // hypersurface case
  else { def m=i*freemodule(n); }       // complete intersection case
//--------------- compute Tjurina module, Tjurina number etc ------------------
  def t1 = jacob(i)+m;                  // Tjurina module/ideal
  def st1 = std(t1);                    // SB of Tjurina module/ideal
  tau = vdim(st1);                      // Tjurina number
  dbprint(printlevel-voice+3,"// Tjurina number = "+string(tau));
  if( size(#)>0 )
  {
     def kB = kbase(st1);               // basis of miniversal deformation
     return(tau,kB,st1,t1);
  }
  return(st1);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r     = 0,(x,y,z),ds;
   poly f     = x5+y6+z7+xyz;        // singularity T[5,6,7]
   list T     = Tjurina(f,"");
   show(T[1]);                       // Tjurina number, should be 16
   show(T[2]);                       // basis of miniversal deformation
   show(T[3]);                       // SB of Tjurina ideal
   show(T[4]); "";                   // Tjurina ideal
   ideal j    = x2+y2+z2,x2+2y2+3z2;
   show(kbase(Tjurina(j)));          // basis of miniversal deformation
   hilb(Tjurina(j));                 // Hilbert series of Tjurina module
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc tjurina (ideal i)
"USAGE:   tjurina(id);  id=ideal or poly
ASSUME:  id=ICIS (isolated complete intersection singularity)
RETURN:  int = Tjurina number of id
NOTE:    Tjurina number = -1 implies that id is not an ICIS
EXAMPLE: example tjurina; shows an example
"
{
   return(vdim(Tjurina(i)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=32003,(x,y,z),(c,ds);
   ideal j=x2+y2+z2,x2+2y2+3z2;
   tjurina(j);
}
///////////////////////////////////////////////////////////////////////////////

proc T_1 (ideal id, list #)
"USAGE:   T_1(id[,<any>]);  id = ideal or poly
RETURN:  T_1(id): of type module/ideal if id is of type ideal/poly.
         We call T_1(id) the T_1-module of id. It is a std basis of the
         presentation of 1st order deformations of P/id, if P is the basering.
         If a second argument is present (of any type) return a list of
         3 modules:
            [1]= T_1(id)
            [2]= generators of normal bundle of id, lifted to P
            [3]= module of relations of [2], lifted to P
                 (note: transpose[3]*[2]=0 mod id)
         The list contains all non-easy objects which must be computed
         to get T_1(id).
DISPLAY: k-dimension of T_1(id) if printlevel >= 0 (default)
NOTE:    T_1(id) itself is usually of minor importance. Nevertheless, from it
         all relevant information can be obtained. The most important are
         probably vdim(T_1(id)); (which computes the Tjurina number),
         hilb(T_1(id)); and kbase(T_1(id)).
         If T_1 is called with two arguments, then matrix([2])*(kbase([1]))
         represents a basis of 1st order semiuniversal deformation of id
         (use proc 'deform', to get this in a direct way).
         For a complete intersection the proc Tjurina is faster.
EXAMPLE: example T_1; shows an example
"
{
   def RR=basering;
   list RRL=ring_list(RR);
   if(RRL[4]!=0)
   {
      int aa=size(#);
      ideal QU=RRL[4];
      RRL[4]=ideal(0);
      def RS=ring(RRL);
      setring RS;
      ideal id=imap(RR,id);
      ideal QU=imap(RR,QU);
      if(aa)
      {
         list RES=T_1(id+QU,1);
      }
      else
      {
         module RES=T_1(id+QU);
      }
      setring RR;
      def RES=imap(RS,RES);
      return(RES);
   }
   ideal J=simplify(id,10);
//--------------------------- hypersurface case -------------------------------
  if( size(J)<2 )
  {
     ideal t1  = std(J+jacob(J[1]));
     module nb = [1]; module pnb;
     dbprint(printlevel-voice+3,"// dim T_1 = "+string(vdim(t1)));
     if( size(#)>0 )
     {
        module st1 = t1*gen(1);
        attrib(st1,"isSB",1);
        return(st1,nb,pnb);
     }
     return(t1);
  }
//--------------------------- presentation of J -------------------------------
   int rk;
   def P = basering;
   module jac, t1;
   jac  = jacob(J);                 // jacobian matrix of J converted to module
   list A=nres(J,2);                // compute presentation of J
   def A(1..2)=A[1..2]; kill A;     // A(2) = 1st syzygy module of J
//---------- go to quotient ring mod J and compute normal bundle --------------
   qring  R    = std(J);
   module jac = fetch(P,jac);
   module t1  = transpose(fetch(P,A(2)));
   list B=nres(t1,2);               // resolve t1, B(2)=(J/J^2)*=normal_bdl
   def B(1..2)=B[1..2]; kill B;
   t1         = modulo(B(2),jac);   // pres. of normal_bdl/trivial_deformations
   rk=nrows(t1);
//-------------------------- pull back to basering ----------------------------
   setring P;
   t1 = fetch(R,t1)+J*freemodule(rk);  // T_1-module, presentation of T_1
   t1 = std(t1);
   dbprint(printlevel-voice+3,"// dim T_1 = "+string(vdim(t1)));
   if( size(#)>0 )
   {
      module B2 = fetch(R,B(2));        // presentation of normal bundle
      list L = t1,B2,A(2);
      attrib(L[1],"isSB",1);
      return(L);
   }
   return(t1);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r     = 32003,(x,y,z),(c,ds);
   ideal i    = xy,xz,yz;
   module T   = T_1(i);
   vdim(T);                      // Tjurina number = dim_K(T_1), should be 3
   list L=T_1(i,"");
   module kB  = kbase(L[1]);
   print(matrix(L[2])*matrix(kB)); // basis of 1st order miniversal deformation
   show(L[2]);                   // presentation of normal bundle
   print(L[3]);                  // relations of i
   print(transpose(matrix(L[3]))*matrix(L[2]));  // should be 0 (mod i)
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc T_2 (ideal id, list #)
"USAGE:   T_2(id[,<any>]);  id = ideal
RETURN:  T_2(id): T_2-module of id . This is a std basis of a presentation of
         the module of obstructions of R=P/id, if P is the basering.
         If a second argument is present (of any type) return a list of
         4 modules and 1 ideal:
            [1]= T_2(id)
            [2]= standard basis of id (ideal)
            [3]= module of relations of id (=1st syzygy module of id) @*
            [4]= presentation of syz/kos
            [5]= relations of Hom_P([3]/kos,R), lifted to P
         The list contains all non-easy objects which must be computed
         to get T_2(id).
DISPLAY: k-dimension of T_2(id) if printlevel >= 0 (default)
NOTE:    The most important information is probably vdim(T_2(id)).
         Use proc miniversal to get equations of the miniversal deformation.
EXAMPLE: example T_2; shows an example
"
{
   def RR=basering;
   list RRL=ring_list(RR);
   if(RRL[4]!=0)
   {
      int aa=size(#);
      ideal QU=RRL[4];
      RRL[4]=ideal(0);
      def RS=ring(RRL);
      setring RS;
      ideal id=imap(RR,id);
      ideal QU=imap(RR,QU);
      if(aa)
      {
         list RES=T_2(id+QU,1);
      }
      else
      {
         module RES=T_2(id+QU);
      }
      setring RR;
      def RES=imap(RS,RES);
      return(RES);
   }

//--------------------------- initialisation ----------------------------------
  def P = basering;
   ideal J = id;
   module kos,SK,B2,t2;
   list L;
   int n,rk;
//------------------- presentation of non-trivial syzygies --------------------
   list A=nres(J,2);                      // resolve J, A(2)=syz
   def A(1..2)=A[1..2]; kill A;
   kos  = koszul(2,J);                    // module of Koszul relations
   SK   = modulo(A(2),kos);               // presentation of syz/kos
   ideal J0 = std(J);                     // standard basis of J
//?*** sollte bei der Berechnung von res mit anfallen, zu aendern!!
//---------------------- fetch to quotient ring mod J -------------------------
   qring R = J0;                          // make P/J the basering
   module A2' = transpose(fetch(P,A(2))); // dual of syz
   module t2  = transpose(fetch(P,SK));   // dual of syz/kos
   list B=nres(t2,2);                     // resolve (syz/kos)*
   def B(1..2)=B[1..2]; kill B;
   t2 = modulo(B(2),A2');                 // presentation of T_2
   rk = nrows(t2);
//---------------------  fetch back to basering -------------------------------
   setring P;
   t2 = fetch(R,t2)+J*freemodule(rk);
   t2 = std(t2);
   dbprint(printlevel-voice+3,"// dim T_2 = "+string(vdim(t2)));
   if( size(#)>0 )
   {
      B2 = fetch(R,B(2));        // generators of Hom_P(syz/kos,R)
      L  = t2,J0,A(2),SK,B2;
      return(L);
   }
   return(t2);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring  r    = 32003,(x,y),(c,dp);
   ideal j    = x6-y4,x6y6,x2y4-x5y2;
   module T   = T_2(j);
   vdim(T);
   hilb(T);"";
   ring r1    = 0,(x,y,z),dp;
   ideal id   = xy,xz,yz;
   list L     = T_2(id,"");
   vdim(L[1]);                           // vdim of T_2
   print(L[3]);                          // syzygy module of id
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////

proc T_12 (ideal i, list #)
"USAGE:   T_12(i[,any]);  i = ideal
RETURN:  T_12(i): list of 2 modules: @*
           *  standard basis of T_1-module =T_1(i), 1st order deformations @*
           *  standard basis of T_2-module =T_2(i), obstructions of R=P/i @*
         If a second argument is present (of any type) return a list of
         9 modules, matrices, integers: @*
             [1]= standard basis of T_1-module
             [2]= standard basis of T_2-module
             [3]= vdim of T_1
             [4]= vdim of T_2
             [5]= matrix, whose cols present infinitesimal deformations @*
             [6]= matrix, whose cols are generators of relations of i(=syz(i)) @*
             [7]= matrix, presenting Hom_P(syz/kos,R), lifted to P @*
             [8]= presentation of T_1-module, no std basis
             [9]= presentation of T_2-module, no std basis
DISPLAY: k-dimension of T_1 and T_2 if printlevel >= 0 (default)
NOTE:    Use proc miniversal from deform.lib to get miniversal deformation of i,
         the list contains all objects used by proc miniversal.
EXAMPLE: example T_12; shows an example
"
{
   def RR=basering;
   list RRL=ring_list(RR);
   if(RRL[4]!=0)
   {
      int aa=size(#);
      ideal QU=RRL[4];
      RRL[4]=ideal(0);
      def RS=ring(RRL);
      setring RS;
      ideal id=imap(RR,id);
      ideal QU=imap(RR,QU);
      if(aa)
      {
         list RES=T_12(id+QU,1);
      }
      else
      {
         list RES=T_12(id+QU);
      }
      setring RR;
      list RES=imap(RS,RES);
      return(RES);
   }

//--------------------------- initialisation ----------------------------------
   int  n,r1,r2,d1,d2;
   def P = basering;
   i = simplify(i,10);
   module jac,t1,t2,sbt1,sbt2;
   matrix Kos,Syz,SK,kbT_1,Sx;
   list L;
   ideal  i0 = std(i);
//-------------------- presentation of non-trivial syzygies -------------------
   list I= nres(i,2);                           // resolve i
   Syz  = matrix(I[2]);                         // syz(i)
   jac = jacob(i);                              // jacobi ideal
   Kos = koszul(2,i);                           // koszul-relations
   SK  = modulo(Syz,Kos);                       // presentation of syz/kos
//--------------------- fetch to quotient ring  mod i -------------------------
   qring   Ox  = i0;                             // make P/i the basering
   module Jac = fetch(P,jac);
   matrix No  = transpose(fetch(P,Syz));        // ker(No) = Hom(syz,Ox)
   module So  = transpose(fetch(P,SK));         // Hom(syz/kos,R)
   list resS  = nres(So,2);
   matrix Sx  = resS[2];
   list resN  = nres(No,2);
   matrix Nx  = resN[2];
   module T_2  = modulo(Sx,No);                  // presentation of T_2
   r2         = nrows(T_2);
   module T_1  = modulo(Nx,Jac);                 // presentation of T_1
   r1         = nrows(T_1);
//------------------------ pull back to basering ------------------------------
   setring P;
   t1   = fetch(Ox,T_1)+i*freemodule(r1);
   t2   = fetch(Ox,T_2)+i*freemodule(r2);
   sbt1 = std(t1);
   d1   = vdim(sbt1);
   sbt2 = std(t2);
   d2   = vdim(sbt2);
   dbprint(printlevel-voice+3,"// dim T_1 = "+string(d1),"// dim T_2 = "+string(d2));
   if  ( size(#)>0)
   {
     if (d1>0)
     {
       kbT_1 = fetch(Ox,Nx)*kbase(sbt1);
     }
     else
     {
       kbT_1 = 0;
     }
     Sx   = fetch(Ox,Sx);
     L = sbt1,sbt2,d1,d2,kbT_1,Syz,Sx,t1,t2;
     return(L);
   }
   L = sbt1,sbt2;
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   int p      = printlevel;
   printlevel = 1;
   ring r     = 199,(x,y,z,u,v),(c,ws(4,3,2,3,4));
   ideal i    = xz-y2,yz2-xu,xv-yzu,yu-z3,z2u-yv,zv-u2;
                            //a cyclic quotient singularity
   list L     = T_12(i,1);
   print(L[5]);             //matrix of infin. deformations
   printlevel = p;
}
///////////////////////////////////////////////////////////////////////////////
proc codim (def id1,def id2)
"USAGE:   codim(id1,id2); id1,id2 ideal or module, both must be standard bases
RETURN:  int, which is:
         1. the vectorspace dimension of id1/id2 if id2 is contained in id1
            and if this number is finite@*
         2. -1 if the dimension of id1/id2 is infinite@*
         3. -2 if id2 is not contained in id1
COMPUTE: consider the Hilbert series iv1(t) of id1 and iv2(t) of id2.
         If codim(id1,id2) is finite,  q(t)=(iv2(t)-iv1(t))/(1-t)^n is
         rational, and the codimension is the sum of the coefficients of q(t)
         (n = dimension of basering).
EXAMPLE: example codim; shows an example
"
{
   if (attrib(id1,"isSB")!=1) { "first argument of codim is not a SB";}
   if (attrib(id2,"isSB")!=1) { "second argument of codim is not a SB";}
   intvec iv1, iv2, iv;
   int i, d1, d2, dd, i1, i2, ia, ie;
  //--------------------------- check id2 < id1 -------------------------------
   ideal led = lead(id1);
   attrib(led, "isSB",1);
   i = size(NF(lead(id2),led));
   if ( i > 0 )
   {
     return(-2);
   }
  //--------------------------- 1. check finiteness ---------------------------
   i1 = dim(id1);
   i2 = dim(id2);
   if (i1 < 0)
   {
     if ( i2 < 0 )
     {
        return(0);
     }
     if (i2 == 0)
     {
       return (vdim(id2));
     }
     else
     {
       return(-1);
     }
   }
   if (i2 != i1)
   {
     return(-1);
   }
   if (i2 <= 0)
   {
     return(vdim(id2)-vdim(id1));
   }
 // if (mult(id2) != mult(id1))
 //{
 //  return(-1);
 // }
  //--------------------------- module ---------------------------------------
   d1 = nrows(module(id1));
   d2 = nrows(module(id2));
   dd = 0;
   if (d1 > d2)
   {
     id2=id2,maxideal(1)*gen(d1);
     dd = -1;
   }
   if (d2 > d1)
   {
     id1=id1,maxideal(1)*gen(d2);
     dd = 1;
   }
  //--------------------------- compute first hilbertseries ------------------
   iv1 = intvec(hilb(id1,1));
   i1 = size(iv1);
   iv2 = intvec(hilb(id2,1));
   i2 = size(iv2);
  //--------------------------- difference of hilbertseries ------------------
   if (i2 > i1)
   {
     for ( i=1; i<=i1; i=i+1)
     {
       iv2[i] = iv2[i]-iv1[i];
     }
     ie = i2;
     iv = iv2;
   }
   else
   {
     for ( i=1; i<=i2; i=i+1)
     {
       iv1[i] = iv2[i]-iv1[i];
     }
     iv = iv1;
     for (ie=i1;ie>=0;ie=ie-1)
     {
       if (ie == 0)
       {
         return(0);
       }
       if (iv[ie] != 0)
       {
         break;
       }
     }
   }
   ia = 1;
   while (iv[ia] == 0) { ia=ia+1; }
  //--------------------------- ia <= nonzeros <= ie -------------------------
   iv1 = iv[ia];
   for(i=ia+1;i<=ie;i=i+1)
   {
     iv1=iv1,iv[i];
   }
  //--------------------------- compute second hilbertseries -----------------
   iv2 = hilb(iv1);
  //--------------------------- check finitenes ------------------------------
   i2 = size(iv2);
   i1 = ie - ia + 1 - i2;
   if (i1 != nvars(basering))
   {
     return(-1);
   }
  //--------------------------- compute result -------------------------------
   i1 = 0;
   for ( i=1; i<=i2; i=i+1)
   {
     i1 = i1 + iv2[i];
   }
   return(i1+dd);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 0,(x,y),dp;
   ideal j = y6,x4;
   ideal m = x,y;
   attrib(m,"isSB",1);  //let Singular know that ideals are a standard basis
   attrib(j,"isSB",1);
   codim(m,j);          // should be 23 (Milnor number -1 of y7-x5)
}

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

proc tangentcone (def id,list #)
"USAGE:   tangentcone(id [,n]); id = ideal, n = int
RETURN:  the tangent cone of id
NOTE:    The procedure works for any monomial ordering.
         If n=0 use std w.r.t. local ordering ds, if n=1 use locstd.
EXAMPLE: example tangentcone; shows an example
"
{
  int ii,n;
  def bas = basering;
  ideal tang;
  if (size(#) !=0) { n= #[1]; }
  if( n==0 )
  {
     def @newr@=changeord(list(list("ds",1:nvars(basering))));
     setring @newr@;
     ideal @id = imap(bas,id);
     @id = std(@id);
     setring bas;
     id = imap(@newr@,@id);
     kill @newr@;
  }
  else
  {
    id = locstd(id);
  }

  for(ii=1; ii<=size(id); ii++)
  {
    tang[ii]=jet(id[ii],mindeg(id[ii]));
  }
  return(tang);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y,z),ds;
   ideal i  = 7xyz+z5,x2+y3+z7,5z5+y5;
   tangentcone(i);
}
///////////////////////////////////////////////////////////////////////////////

proc locstd (def id)
"USAGE:   locstd (id); id = ideal
RETURN:  a standard basis for a local degree ordering
NOTE:    the procedure homogenizes id w.r.t. a new 1st variable @t@, computes
         a SB w.r.t. (dp(1),dp) and substitutes @t@ by 1.
         Hence the result is a SB with respect to an ordering which sorts
         first w.r.t. the order and then refines it with dp. This is a
         local degree ordering.
         This is done in order to avoid cancellation of units and thus
         be able to use option(contentSB);
EXAMPLE: example locstd; shows an example
"
{
  int ii;
  def bas = basering;
  ring @r_locstd = create_ring(ring_list(bas)[1], "(@t@,"+varstr(bas)+")", "(dp(1),dp)", "no_minpoly");
  ideal @id = imap(bas,id);
  ideal @hid = homog(@id,@t@);
  @hid = std(@hid);
  @hid = subst(@hid,@t@,1);
  setring bas;
  def @hid = imap(@r_locstd,@hid);
  attrib(@hid,"isSB",1);
  kill @r_locstd;
  return(@hid);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y,z),ds;
   ideal i  = xyz+z5,2x2+y3+z7,3z5+y5;
   locstd(i);
}