source: git/Singular/LIB/classifyMapGerms.lib @ 20f2239

///////////////////////////////////////////////////////////////////////////////
version="version classifyMapGerms.lib 4.1.2.0 Feb_2019 "; // $Id$
category="Singularities";
info="
LIBRARY:   classifyMapGerms.lib
AUTHORS:   Gerhard Pfister, pfister@mathematik.uni-kl.de
           Deeba Afzal,     deebafzal@gmail.com
           Shamsa Kanwal,   lotus_zone16@yahoo.com

OVERVIEW:
 A library for computing the standard basis of the tangent space at the orbit
 of an algebraic group action. The tangent space is usually described as the sum
 of two modules over different rings. It computes the standard basis using
 modular methods and parallel modular methods. It also computes the normal form
 of the germ given by Riegers classification.

REFERENCES:
[1] Idrees N.; Pfister, G.; Steidel, S.: Parallelization of modular algorithms.
  J. Symbolic Comput. 46(2011), no. 6, 672-684.
[2] Gibson,C.G; Hobbs,C.A.: Simple SIngularities of Space Curves.
  Math.Proc. Comb.Phil.Soc.(1993),113,297.
[3] Bruce, J.W.,Gaffney, T.J.: Simple singularities of mappings (C, 0) ->(C2,0).
  J. London Math. Soc. (2) 26 (1982), 465-474.
[4] Rieger, J. H.: Families of maps from the plane to the plane. J. London Math.
Soc. (2)36(1987), no. 2. 351-369.

PROCEDURES:
 coDimMap(I);    computes a bound of the A-determinacy of the map germ defined by I
 coDim(M,N,I,b); computes the K-vectorspace dimension of A^r/M+N+maxideal(b)*A^r
 vStd(M,N,I,b);      computes a standard basis of M+N+maxideal(b)*A^r
 modVStd(M,N,I,b);   computes a standard basis of M+N+maxideal(bound)*A^r (modular)
 modVStd0(M,N,I,b);  computes a standard basis of M+N+maxideal(bound)*A^r (parallel)
 classifySimpleMaps(I);  computes the normal form of a germ in Riegers classification
 classifySimpleMaps1(I); computes the normal form of a germ in Riegers classification
 classifyUnimodalMaps(I); computes the normal form of a germ in Riegers classification
";

LIB "general.lib";
LIB "modstd.lib";
LIB "teachstd.lib";
LIB "algebra.lib";

//===================================================================================
proc coDimMap(ideal I, list #)
"USAGE:   coDimMap(I, #);  I=ideal, #=list
COMPUTE:  a bound of the A-determinacy of the map germ defined by I.
RETURN:   a list in which 1st entry gives the bound of the A-determinancy and
          the second entry gives the codimension of the map germ defined by I.
NOTE:     if # is empty it computes A^e-codimension(the extended codimension).
EXAMPLE:  example coDimMap; shows an example"
{
   int bound=10;
   module M=jacob(I);
   module N=freemodule(ncols(I));
   if(size(#)==0){return(coDim(M,N,I,bound,0));}
   M=maxideal(1)*M;
   N=I*N;
   list L=coDim(M,N,I,bound,0);
   L[2]=L[2]-nvars(basering);
   return(L);
}
example
{
 "EXAMPLE"; echo=2;
  ring R=0,(x,y),(c,ds);
  poly f1=x;
  poly f2=xy+y5+y7;
  poly f11=f1+f2*f1;
  poly f22=f2+f1^2;
  map phi=basering,x+y,y+y2;
  f1=phi(f11);
  f2=phi(f22);
  ideal I=f1,f2;
  coDimMap(I);
  coDimMap(I,1);
}
///////////////////////////////////////////////////////////////////////////////
proc coDim(module M, module N, ideal I,int bound, list #)
"USAGE:   coDim(module M, module N, ideal I,int bound, list #); M is a submodule
          in A^r over the basering=:A, N is a submodule in R^r over the subring
          R of the basering generated by the entries of I
COMPUTE:  computes the K-vectorspace dimension of A^r/M+N+maxideal(bound)*A^r
RETURN:   an integer
NOTE:     if # is not empty the bound is corrected by computing determinacy
EXAMPLE:  example coDim; shows an example"
{
//---------------------------- initialisation ---------------------------------
   int r=nrows(M);
   int k;
   int oldbound=bound;
   module M0=M;
   module N0=N;
   if(size(#)!=0)
   {
      if(#[1]==0)
      {
         k=boundK(M,N,I);
         if(k<0){return(list(-1,-1));}
      }
      else
      {
         k=#[1];
      }
      bound=bound+k;
   }
   list S=vStd(M,N,I,bound);
   M=S[1];
   module L=S[2];
   L=simplify(reduce(lead(L),std(lead(M)+maxideal(bound)*freemodule(r))),1);
   L=simplify(L,6);
   if(size(#)==0){
   return(vdim(std(lead(M)+maxideal(bound)*freemodule(r)))-size(L));
   }
   if(correctBound(M,L,bound,oldbound))
   {
      int co=vdim(std(lead(M)+maxideal(bound)*freemodule(r)))-size(L);
      return(list(bound,co));
   }
   bound++;
   return(coDim(M0,N0,I,bound,k));
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   module M=maxideal(1)*jacob(I);
   module N=I*freemodule(2);
   coDim(M,N,I,15);
}
///////////////////////////////////////////////////////////////////////////////
proc vStd(module M, module N, ideal I,int bound)
"USAGE:    vStd(M, N, I, bound);M is a submodule  in A^r over the basering=:A,
           N ist a submodule in R^r over the subring R of the basering generated
           by the entries of I
COMPUTE:  a standard basis of M+N+maxideal(bound)*A^r
RETURN:   a list whose ist entry gives a list where each entry is a  genrator of
          standard basis and second entry gives a list of generators after the
          reduced echelon form
EXAMPLE:  example vStd; shows an example"
{
   if(size(M)!=1)
   {
      M=jet(std(M+maxideal(bound)*freemodule(nrows(M))),bound);
   }
   if (system("version")>4033)
   {
     N=system("reduce_bound",N,M,bound);
   }
   else
   {
     N=myReduceM(N,M,bound);
   }
   N=echelon(M,N,bound);
   module L=computeN(M,N,I,bound-1);
   L=echelon(M,L,bound);
   return(list(M,L));
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   module M=maxideal(1)*jacob(I);
   module N=I*freemodule(2);
   vStd(M,N,I,15);
}
///////////////////////////////////////////////////////////////////////////////
static proc vStd_size1(module M, module N, ideal I, int bound)
"USAGE:    vStd(M, N, I, bound);M is a submodule generated by one element in A^r over the
           basering=:A, N ist a submodule in R^r over the subring R of the basering
           generated by the entries of I
COMPUTE:  a standard basis of M+N+maxideal(bound)*A^r
RETURN:   a list whose ist entry gives a list where each entry is a  genrator of
          standard basis and second entry gives a list of generators after the
          reduced echelon form
EXAMPLE:  example vStd_size1; shows an example"
{
   if(size(M)!=1)
   {
      M=jet(std(M+maxideal(bound)*freemodule(nrows(M))),bound);
   }
   if (system("version")>4033)
   {
     N=system("reduce_bound",N,M,bound);
   }
   else
   {
     N=myReduceM(N,M,bound);
   }
   N=echelon(M,N,bound);
   module L=computeN(M,N,I,bound-1);
   L=echelon(M,L,bound);
   return(L);
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   module M=maxideal(1)*jacob(I);
   module N=I*freemodule(2);
   vStd_size1(M,N,I,15);
}

///////////////////////////////////////////////////////////////////////////////
static proc boundK(module M,module N, ideal I)
"USAGE:    boundK(M, N, I);
COMPUTE:  Let T=freemodule(nrows(M)), computes k such that
          maxideal(k)*T<=M+I*T+maxideal(k+1)*T
RETURN:   an integer
REMARk:   needed to compute the determinacy
"
{
   int i;
   module T=freemodule(nrows(M));
   module M0=M+I*T;
   while(1)
   {
      i++;
      T=maxideal(1)*T;

      M=std(jet(M0,i)+maxideal(1)*T);
      if(size(reduce(T,M,5))==0){return(i);}
      if(i>100){M0=std(M0);if(vdim(M0)==-1){break;}}
   }
   return(-1);
}
///////////////////////////////////////////////////////////////////////////////
static proc correctBound(module M,module L,int bound,int oldbound)
"USAGE:     correctBound(M, L, bound, oldbound)
COMPUTE:    let T=freemodule(nrows(M)),test if
            maxideal(old bound)*T<=M+L+maxideal(bound)*T
RETURN:     0 or 1
REMARK:     needed to compute the A-determinacy and codimension
"
{
   int i;
   L=jet(L+std(lead(M)),bound-1);
   module T=maxideal(oldbound)*freemodule(nrows(M));
   while(i<bound-oldbound)
   {
      T=jet(myReduceM(T,M,bound),bound-1);
      if(size(reduceV(M,T,L,bound))!=0)
      {
         return(0);
      }
      T=maxideal(1)*T;
      i++;
   }
   return(1);
}
///////////////////////////////////////////////////////////////////////////////
static proc reduceV(module M,module T, module L,int bound)
"USAGE:     reduceV(M, T, L, bound)
COMPUTE:    special reduction procedure for correctBound (reduction modulo a sum
            of modules over different rings
RETURN:     a list
"
{
   int i,j;
   vector f,g;
   for(i=1;i<=ncols(T);i++)
   {
     f=T[i];
     while(1)
     {
        g=reducer(f,L);
        if(g==0){break;}
        f=f-leadcoef(f)/leadcoef(g)*g;
        //f=jet(system("reduce_bound",f,M,bound),bound-1);
        f=jet(myReduce(f,M,bound),bound-1);
        if(f==0){break;}
     }
     T[i]=f;
   }
   return(T);
}
///////////////////////////////////////////////////////////////////////////////
static proc reducer(vector f, module L)
"USAGE:     reducer(f, module L)
COMPUTE:    find the reductor for reduceV
RETURN:     a vector
"
{
   vector g;
   int i;
   for(i=1;i<=size(L);i++)
   {
      if(leadmonom(f)==leadmonom(L[i]))
      {
         g=L[i];
         break;
      }
   }
   return(g);
}
///////////////////////////////////////////////////////////////////////////////
static proc computeN(module M, module N, ideal I, int bound)
"USAGE:   computeN(M, N, I, bound) M is a submodule  in A^r over the basering=:A
          N ist a submodule in R^r over the subring R of the basering
          generated by the entries of I
COMPUTE:  all products of powers of generators of I with N
"
{
   I=jet(I,bound);
   I=simplify(I,2);
   module L;
   int j,k;
   int r=size(I);
   ideal F=1;
   ideal G;

   while(F!=0)
   {
      for(k=1;k<=size(N);k++)
      {
         for(j=1;j<=size(F);j++)
         {
            L[size(L)+1]=jet(myReduce(F[j]*N[k],M,bound),bound);
            //L[ncols(L)+1]=jet(system("reduce_bound",F[j]*N[k],M,bound),bound);
         }
      }
      G=F;
      F=0;
      for(j=1;j<=size(G);j++)
      {
         for(k=1;k<=r;k++)
         {
            F=F,jet(jet(G[j],bound-ord(I[k]))*I[k],bound);
         }
      }
      F=simplify(F,2);
      F=simplify(F,4);
      F=simplify(F,2);
   }
   return(L);
}

///////////////////////////////////////////////////////////////////////////////
static proc myReduceM(module N, module M, int bound)
"USAGE:   myReduceM(N, M, bound) M is a submodule  in A^r over the basering=:A
          N ist a submodule in R^r over the subring R of the basering
          generated by the entries of I
COMPUTE:  myReduce for ideals
"
{
   int i;
   for(i=1;i<=ncols(N);i++)
   {
      N[i]=myReduce(N[i],M,bound);
   }
   return(N);
}
///////////////////////////////////////////////////////////////////////////////
static proc myReduce(vector f, module M, int bound)
{
   f=jet(f,bound);
   if(f==0){return(f);}
   vector g, vv,ww;
   poly p;
   number c;
   int i;
   vector re;
   while(f!=0)
   {
      for(i=1;i<=size(M);i++)
      {
         if(reduce(lead(f),std(lead(M[i])),5)==0)
         {
            c=leadcoef(f)/leadcoef(M[i]);
            vv=lead(f);ww=lead(M[i]);
            p=quotient(vv,ww)[1];
            g=M[i];break;
         }
      }
      while((f!=0)&&(g!=0))
      {
         f=f-c*p*g;
         f=jet(f,bound);
         if(f==0){break;}
         g=0;
         for(i=1;i<=size(M);i++)
         {
            if(reduce(lead(f),std(lead(M[i])),5)==0)
            {
               c=leadcoef(f)/leadcoef(M[i]);
               vv=lead(f);ww=lead(M[i]);
               p=quotient(vv,ww)[1];
               g=M[i];break;
            }
         }
      }
      re=re+lead(f);
      f=f-lead(f);
   }
   return(re);
}
///////////////////////////////////////////////////////////////////////////////
static proc echelon(module M, module N, int b)
"USAGE:   echelon(M, N, b)
COMPUTE:  the echelon form of N modulo M
RETURN:   the echelon form of N modulo M
"
{
   int i,j,d;
   N=simplify(N,4);
   N=simplify(N,3);
   N=sort(N)[1];
   d=size(N);
   i=d;
   while(i>=2)
   {
      for(j=i-1;j>=1;j=j-1)
      {
         if(lead(N[i])==lead(N[j]))
         {
            N[j]=N[j]-N[i];

         }
      }
      N=simplify(N,4);
      N=simplify(N,3);
      N=sort(N)[1];
      i=i-(d-size(N))-1;
      d=size(N);
   }
   N=simplify(N,6);
   return(N);
}
///////////////////////////////////////////////////////////////////////////////
static proc reduc(poly f,ideal I, int b)
"USAGE:   reduc(f, I, b)
COMPUTE:  the normal form of f with respect to the algebra generated by I
"
{
   list L=1;
   map psi;
   f=jet(f,b);
   if(f==0){return(f);}
   while((f!=0) && (L[1]!=0))
   {
     L= algebra_containment(lead(f),lead(I),1);
     if (L[1]==1)
     {
        def S= L[2];
        psi= S,maxideal(1),I;
        f=jet(f-psi(check),b);
        kill S;
     }
   }
   return (lead(f)+reduc(f-lead(f),I,b));
}
///////////////////////////////////////////////////////////////////////////////
proc classifySimpleMaps(ideal I)
"USAGE:   classifySimpleMaps(I);  I=an ideal with 2 generators in a polynomial
         ring with 2 variables and local ordering defining a map germ C^2 to C^2
COMPUTE:  The normal form of the germ in Riegers classification if it is
          simple
RETURN:   normal form of I, of type ideal
NOTE:     If I is not simple it returns (0,0)
EXAMPLE:  example classifySimpleMaps; shows an example"
{
   if(size(jet(I,1))==0){return(ideal(0,0));}
   list L=coDimMap(I,1);
   int determinacy=L[1];
   int codimension=L[2];
   if(determinacy<0){return(ideal(0,0));}
   I=normalMap(I,determinacy);
   poly p=lead(I[2]);
   if(p==0){return(ideal(0,0));}
   if(p==var(2)){return(maxideal(1));}
   if(p==var(2)^2){return(ideal(var(1),var(2)^2));}
   if(p==var(1)*var(2))
   {
      if(codimension==2){return(ideal(var(1),p+var(2)^3));}
      if(codimension==3){return(ideal(var(1),p+var(2)^4));}
      if(codimension==4){return(ideal(var(1),p+var(2)^5+var(2)^7));}
      if(codimension==5){return(ideal(var(1),p+var(2)^5));}
      return(ideal(0,0));
   }
   if(p==var(2)^3)
   {
      return(ideal(var(1),p+var(1)^(codimension-1)*var(2)));
   }
   if(p==var(1)*var(2)^2)
   {
      if(vdim(std(I))==4)
      {
         return(ideal(var(1),p+var(2)^4+var(2)^(2*codimension-3)));
      }
      if(codimension==5){return(ideal(var(1),p+var(2)^5+var(2)^6));}
      if(codimension==6){return(ideal(var(1),p+var(2)^5+var(2)^9));}
      if(codimension==7){return(ideal(var(1),p+var(2)^5));}
      return(ideal(0,0));
   }
   if(p==var(1)^2*var(2))
   {
      if(codimension==3){return(ideal(var(1),p+var(2)^3));}
      if(codimension==5){return(ideal(var(1),p+var(2)^4+var(2)^5));}
      if(codimension==6){return(ideal(var(1),p+var(2)^4));}
      return(ideal(0,0));
   }
   return(ideal(0,0));
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   classifySimpleMaps(I);
}
///////////////////////////////////////////////////////////////////////////////
proc classifySimpleMaps1(ideal I)
"USAGE:   classifySimpleMaps1(I);  I=an ideal with 2 generators in a polynomial
         ring with 2 variables and local ordering defining a map germ C^2 to C^2
COMPUTE:  The normal form of the germ in Riegers classification if it is
          simple
RETURN:   normal form of I, of type ideal
NOTE:     If I is not simple it returns (0,0)
EXAMPLE:  example classifySimpleMaps1; shows an example"
{
   if(size(jet(I,1))==0){return(ideal(0,0));}
   list L=coDimMap(I,1);
   int determinacy=L[1];
   int c=L[2];
   if(determinacy<0){return(ideal(0,0));}
   I=normalMap(I,determinacy);
   int m =vdim(std(I));
   int mu=vdim(std(jacob(diff(I[2],var(2)))));

   if((mu==-1)&&(m>=2)){return(ideal(0,0));}
   if(m==1){return(maxideal(1));}

   if(mu==0)
   {
      if(m==2){return(ideal(var(1),var(2)^2));}
      if(m==3){return(ideal(var(1),var(1)*var(2)+var(2)^3));}
      if(m==4){return(ideal(var(1),var(1)*var(2)+var(2)^4));}
      if(m==5)
      {
        if(c==4)
        {
           return(ideal(var(1),var(1)*var(2)+var(2)^5+var(2)^7));
        }
        if(c==5)
        {
           return(ideal(var(1),var(1)*var(2)+var(2)^5));
        }
      }
      return(ideal(0,0));
   }
   if(mu==1)
   {
      if(m==3){return(ideal(var(1),var(1)^2*var(2)+var(2)^3));}
      if(m==4){return(ideal(var(1),var(1)*var(2)^2+var(2)^4+var(2)^(2*c-3)));}
      if(m==5)
      {
        if(c==5)
        {
           return(ideal(var(1),var(1)*var(2)^2+var(2)^5+var(2)^6));
        }
        if(c==6)
        {
           return(ideal(var(1),var(1)*var(2)^2+var(2)^5+var(2)^9));
        }
        if(c==7)
        {
           return(ideal(var(1),var(1)*var(2)^2+var(2)^5));
        }
      }
      return(ideal(0,0));
    }
   if(mu==2)
   {
      if(m==3){return(ideal(var(1),var(2)^3+var(1)^3*var(2)));}
      if(m==4)
      {
        if(c==5)
        {
           return(ideal(var(1),var(1)^2*var(2)+var(2)^4+var(2)^5));
        }
        if(c==6)
        {
           return(ideal(var(1),var(1)^2*var(2)+var(2)^4));
        }
      }
      return(ideal(0,0));
    }
   if(mu>2)
   {
      if(m==3)
      {
         return(ideal(var(1),var(2)^3+var(1)^(c+1)*var(2)));
      }
      return(ideal(0,0));
   }
   return(ideal(0,0));
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   classifySimpleMaps1(I);
}
///////////////////////////////////////////////////////////////////////////////
proc modVStd0(module M, module N, ideal I, int bound)
"USAGE:   modVStd0((M, N, I,bound, #); M is a submodule
          in A^r over the basering=:A, N ist a submodule in R^r over the subring
          R of the basering generated by the entries of I
COMPUTE:  a standard basis of M+N+maxideal(bound)*A^r using the parallel modular
          version
RETURN:   a list whose ist entry gives a list where each entry is a  genrator of
          standard basis and second entry gives a list of generators after the
          reduced echelon form
NOTE:     if # is not empty the bound is corrected by computing determinacy
EXAMPLE:  example modVStd0; shows an example"
{
    /* call modular() */
    if (size(M) > 1) {
        list W = modular("vStd", list(M, N, I, bound),
            Modular::primeTest_default, deleteUnluckyPrimesM0,
            pTestSBM0, finalTestM0);
        attrib(W[1], "isSB", 1);
    }
    else {
        module L = modular("vStd_size1", list(M, N, I, bound),
            Modular::primeTest_default, Modstd::deleteUnluckyPrimes_std,
            pTestSBM0_size1, finalTestM0_size1);
        attrib(M, "isSB", 1);
        list W = list(M, L);
    }

    /* return the result */
    return(W);
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   module M=maxideal(1)*jacob(I);
   module N=I*freemodule(2);
   modVStd0(M,N,I,15);
}
///////////////////////////////////////////////////////////////////////////////
proc modVStd(module M, module N, ideal I,int bound,list #)
"USAGE:   modVStd((M, N, I,bound, #); M is a submodule
          in A^r over the basering=:A, N ist a submodule in R^r over the subring
          R of the basering generated by the entries of I
COMPUTE:  a standard basis of A^r/M+N+maxideal(bound)*A^r using modular version
RETURN:   a list whose ist entry gives a list where each entry is a  genrator of
          standard basis and second entry gives a list of generators after the
          reduced echelon form
NOTE:     if # is not empty the bound is corrected by computing determinacy
EXAMPLE:  example modVStd; shows an example"
{
//---------------------------- initialisation --------------------------------
   int np=10;
   bigint H;
   if(size(#)>0){np=#[1];}
   list P=2134567879;
   int j,cc;
   def R0 = basering;
   module L;
   module M0=M;
   module N0=N;
   list T1,T2,L1,L2;
   list rl = ringlist(R0);

   while(1)
   {
      cc=timer;
      for(j = 1; j <= np; j++)
      {
         P[size(P)+1]=prime(P[size(P)]-1);
      }
      for(j = 1; j <= size(P); j++)
      {
         rl[1] = P[j];
         def @r = ring(rl);
         setring @r;
         ideal I  = fetch(R0,I);
         module M = fetch(R0,M0);
         module N = fetch(R0,N0);
         list W = vStd(M,N,I,bound);
         setring R0;
         list W = fetch(@r,W);
         T1[j]=W[1];
         T2[j]=W[2];
         kill W;
         kill @r;
      }
      L1 = deleteUnluckyPrimesM(T1,P);
      L2 = deleteUnluckyPrimesM(T2,P);
      if(size(M0)>1)
      {
         H=1;
         for(j = 1; j <= size(L1[2]); j++)
         {
            H = H*L1[2][j];
         }
         M=  chinrem(L1[1],L1[2]);
         M = farey(M,H);
         attrib(M,"isSB",1);
      }
      else
      {
         M=M0;
      }
      H=1;
      for(j = 1; j <= size(L2[2]); j++)
      {
         H = H*L2[2][j];
      }
      L=  chinrem(L2[1],L2[2]);
      L = farey(L,H);

      if(pTestSBM(M0,N0,I,M,L,P,bound))
      {
         //int bb=timer;
         if(finalTestM(M0,N0,I,M,L,bound))
         {
          // "time for final test";timer-bb;
           return(list(M,L));
         }
         // "time for final test";timer-bb;
        // "final test failed";
      }
     // "pTest failed";size(P);
   }
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y5+y7;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   module M=maxideal(1)*jacob(I);
   module N=I*freemodule(2);
   modVStd(M,N,I,15);
}
///////////////////////////////////////////////////////////////////////////////
static proc deleteUnluckyPrimesM(list T, list L)
{
   int j,k,c;
   intvec hl,hc;
   module cT,lT,cK;
   lT = lead(T[size(T)]);
   attrib(lT,"isSB",1);
      for(j = 1; j < size(T); j++)
      {
         cT = lead(T[j]);
         attrib(cT,"isSB",1);
         if(size(reduce(cT,lT,5))!=0)
         {
            cK = cT;
            c++;
         }
         else
         {
           if(size(reduce(lT,cT,5))!=0)
           {
              cK = cT;
              c++;
           }
         }
      }
      if(c > size(T) div 2){ lT = cK; }

   int addList;
   j = 1;
   attrib(lT,"isSB",1);
   while((j <= size(T))&&(c > 0))
   {
      cT = lead(T[j]);
      attrib(cT,"isSB",1);
      if((size(reduce(cT,lT,5)) != 0)||(size(reduce(lT,cT,5)) != 0))
      {
         T = delete(T,j);
         if(j == 1)
         {
            L = L[2..size(L)];
            if(addList == 1) { M = M[2..size(M)]; }
         }
         else
         {
            if(j == size(L))
            {
               L = L[1..size(L)-1];
               if(addList == 1) { M = M[1..size(M)-1]; }
            }
            else
            {
               L = L[1..j-1],L[j+1..size(L)];
               if(addList == 1) { M = M[1..j-1],M[j+1..size(M)]; }
            }
         }
         j--;
      }
      j++;
   }

   for(j = 1; j <= size(L); j++)
   {
      L[j] = bigint(L[j]);
   }

   if(addList == 0) { return(list(T,L,lT)); }
   if(addList == 1) { return(list(T,L,M,lT)); }
}

static proc deleteUnluckyPrimesM0(alias list modresults)
{
    list T1, T2;
    int i;
    for (i = size(modresults); i > 0; i--) {
        T1[i] = modresults[i][1];
        T2[i] = modresults[i][2];
    }
    list indices1 = Modstd::deleteUnluckyPrimes_std(T1);
    list indices2 = Modstd::deleteUnluckyPrimes_std(T2);
    return(indices1+indices2);
}

///////////////////////////////////////////////////////////////////////////////
static proc pTestSBM(module M0, module N0, ideal I, module M, module L, list P,
int bound)
{
   int i,j,k,p;
   def R0 = basering;
   list r = ringlist(R0);

   while(!j)
   {
      j = 1;
      p = prime(random(1000000000,2134567879));
      for(i = 1; i <= size(P); i++)
      {
         if(p == P[i]) { j = 0; break; }
      }

   }
   r[1] = p;
   def @R = ring(r);
   setring @R;
   ideal I  = fetch(R0,I);
   module M0 = fetch(R0,M0);
   module N0 = fetch(R0,N0);
   list W = vStd(M0,N0,I,bound);
   module M = fetch(R0,M);
   module L = fetch(R0,L);
   attrib(M,"isSB",1);
   attrib(L,"isSB",1);
   j=size(reduce(W[1],M,5))+size(reduce(W[2],L,5));
   setring R0;
   return(!j);
}
///////////////////////////////////////////////////////////////////////////////
static proc pTestSBM0(string command, alias list args, def result, int p)
{
    def R0 = basering;
    list r = ringlist(R0);
    r[1] = p;
    def @R = ring(r);
    setring @R;
    list args = fetch(R0, args);
    list W = vStd(args[1..4]);
    def result = fetch(R0, result);
    attrib(result[1],"isSB",1);
    attrib(result[2],"isSB",1);
    int j=size(reduce(W[1],result[1],5))+size(reduce(W[2],result[2],5));
    setring R0;
    return(!j);
}
///////////////////////////////////////////////////////////////////////////////

static proc pTestSBM0_size1(string command, alias list args, def result, int p)
{
    // adjusted copy of pTestSBM0() for the case size(args[1]) <= 1
    def R0 = basering;
    list r = ringlist(R0);
    r[1] = p;
    def @R = ring(r);
    setring @R;
    list args = fetch(R0, args);
    module L = vStd_size1(args[1..4]);
    def result = fetch(R0, result);
    attrib(result,"isSB",1);
    int j = size(reduce(L, result,5));
    setring R0;

    return(!j);
}
///////////////////////////////////////////////////////////////////////////////
static proc finalTestM(module M0, module N0, ideal I, module M, module L,
int bound)
{
   attrib(M,"isSB",1);
   int a=size(myReduceM(M0,M,bound));
   int b=testSBM(M,bound);
   int c=size(reduceV(M,myReduceM(N0,M,bound),L,bound));
   return(!(a+b+c));
}
///////////////////////////////////////////////////////////////////////////////
static proc testSBM(module M, int bound)
{
   int i,j;
   vector f;
   for(i=1;i<size(M);i++)
   {
      for(j=i+1;j<=size(M);j++)
      {
         f=jet(spoly(M[i],M[j],1),bound);
         f=myReduce(f,M,bound);
         if(f!=0){return(1);}
      }
   }
   return(0);
}
///////////////////////////////////////////////////////////////////////////////
static proc finalTestM0(string command, alias list args, def result)
{
    attrib(result[1], "isSB", 1);
    if (size(myReduceM(args[1], result[1], args[4]))) {
        return(0);
    }
    if(testSBM(result[1],args[4])){return(0);}
    if (size(reduceV(result[1], myReduceM(args[2], result[1], args[4]),
        result[2], args[4]))) {
        return(0);
    }
    return(1);
}
///////////////////////////////////////////////////////////////////////////////
static proc finalTestM0_size1(string command, list args, def result)
{
    attrib(args[1], "isSB", 1);
    if (size(reduceV(args[1], myReduceM(args[2], args[1], args[4]),
        result, args[4]))) {
        return(0);
    }
    return(1);
}
///////////////////////////////////////////////////////////////////////////////
static proc normalMap(ideal I, int bound)
"USAGE:    normalMap(I);  I=f1,f2 in K[x,y]
 RETURN:   (0,0) if jet(I,1)=0
           (x,h) A-equivalent to I modulo <x,y>^bound+1
           if ord(h)=1 then h=y
           if ord(h)=2 then h=xy+y^m(I)+h.o.t.if m(I)>=3 or (x,y^2)
           if ord(h)=3 milnor(diff(h,y))=1 and m(I)>3 then h=xy2+y^m(I)+h.o.t.
           if ord(h)=4 and m(I)=4 then h=y^4+a*x^3y+b*x^2y^2+h.o.t.
           m(I)=vdim(std(I))
 NOTE: the procedure is designed for simple and unimodular maps
"
{
   I=jet(I,bound);
   //the trivial cases
   if(size(jet(I,1))==0){return(ideal(0,0));}
   if(vdim(std(jet(I,1)))==1){return(maxideal(1));}
   if(vdim(std(I))==2){I=var(1),var(2)^2;return(I);}

   def R=basering;
   map phi;
   poly p;
   int i;

   //transformation to (x,h)
   if(jet(I[1],1)==0){p=I[1];I[1]=I[2];I[2]=p;}
   I[1]=simplify(I[1],1);
   if(lead(I[1])==var(2)){phi=R,var(2),var(1);}
   else{phi=R,var(1)-I[1][2],var(2);}
   I=phi(I);
   while(I[1]!=var(1))
   {
      phi=R,2*var(1)-I[1],var(2);
      I=jet(phi(I),bound);
   }
   while((leadmonom(I[2])==var(1))||(leadmonom(I[2])==var(1)^2)||(leadmonom(I[2]
   )==var(1)^3)||(leadmonom(I[2])==var(1)^4))
   {
      I[2]=I[2]-lead(I[2]);
   }
   I[2]=simplify(I[2],1);
   if(deg(lead(I[2]))>=5){return(I);}

   //special treatment of h
   //the case ord(h)=4
   if(deg(lead(I[2]))==4)
   {
      if(vdim(std(I))>4){return(I);}
      def S=changevar("var(2),var(1)",R);
      setring S;
      map psi=R,var(2),var(1);
      ideal I=psi(I);
      I[2]=simplify(I[2],1);
      if(leadmonom(I[2][2])==var(1)^3*var(2))
      {
         map sigma=S,var(1)-1/4*leadcoef(I[2][2])*var(2),var(2);
         I=sigma(I);
         setring R;
         map lambda=S,var(2),var(1);
         I=lambda(I);
         if(leadmonom(I[2])==var(1)^4){I[2]=I[2]-lead(I[2]);}
         return(I);
      }
      else
      {
         setring R;
         return(I);
      }
   }

   //the case ord(h)=3
   if(deg(lead(I[2]))==3)
   {
      phi=R,var(1),var(2)-leadcoef(I[2])/(2*leadcoef(I[2][2]))*var(1);
      I=phi(I);
   }
   while((leadmonom(I[2])==var(1))||(leadmonom(I[2])==var(1)^2)||(leadmonom(I[2]
   )==var(1)^3)||(leadmonom(I[2])==var(1)^4))
   {
      I[2]=I[2]-lead(I[2]);
   }
   I[2]=simplify(I[2],1);
   if(leadmonom(I[2])==var(1)*var(2)^2)
   {
      while(i<bound)
      {
         i++;
         p=jet(I[2]/(var(1)*var(2))-var(2),i);
         phi=R,var(1),var(2)-1/2*p;
         I=jet(phi(I),bound);
      }
   }

   //the case ord(h)=2
   if(leadmonom(I[2])==var(1)*var(2))
   {
      while(i<bound)
      {
         i++;
         p=jet(I[2]/var(1)-var(2),i);
         phi=R,var(1),var(2)-p;
         I=jet(phi(I),bound);
      }
   }

   //all cases
   p=lead(I[2])+reduc(I[2]-lead(I[2]),I,bound);
   I[2]=p;
   if((leadmonom(I[2])==var(1)*var(2))||(leadmonom(I[2])==var(1)*var(2)^2))
   {
      number a=leadcoef(I[2][2]);
      I=subst(I,x,a*x);
      I=1/a*I;
   }
   return(I);
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   ideal I1=x,xy2+y6+y7+7y9;
   ideal I2=x,xy+y6+y8+77y9;
   ideal I3=x,3x4+2x3y+7x2y2+4xy3+11y4;
   map phi=R,x+7y+y2,2x+y+y3;
   ideal J=phi(I1);
   normalMap(J,10);
   J=phi(I2);
   normalMap(J,10);
   J=I3;
   normalMap(J,10);
}
///////////////////////////////////////////////////////////////////////////////
proc classifyUnimodalMaps(ideal I)
"USAGE:   classifyUnimodalMaps(I);  I an ideal with 2 generators in a polynomial
         ring with 2 variables and local ordering defining a map germ C^2 to C^2
COMPUTE:  The normal form of the germ in Riegers classification if it is
          simple
RETURN:   normal form of I, of type ideal
NOTE:     If I is not unimodal  it returns (0,0)
EXAMPLE:  example classifyUnimodalMaps; shows an example"
{
   if(size(jet(I,1))==0){return(ideal(0,0));}
   list L=coDimMap(I,1);
   int determinacy=L[1];
   int c=L[2];
   if(determinacy<0){return(ideal(0,0));}
   I=normalMap(I,determinacy);

   poly g=I[2];
   int m =vdim(std(I));
   int mu=vdim(std(jacob(diff(g,var(2)))));
   int d=doubleFoldNumber(g);

   if((mu==-1)||((m!=4)&&(m!=6))||(deg(lead(I[2]))>4)){return(ideal(0,0));}

   if(m==6)
   {
      if((mu==0))
      {
         if(c==6)
         {
            g=jet(g,9);
            I[2]=modulus(g);
            return(I);
         }
         if(c==7){return(ideal(var(1),var(1)*var(2)+var(2)^6+var(2)^14));}
         if(c==8){return(ideal(var(1),var(1)*var(2)+var(2)^6));}
         return(ideal(0,0));
      }
      if((mu==1)&&(c==7))
      {
         g=jet(g,9);
         I[2]=modulus(g);
         return(I);
      }
   }
   if((mu==4))
   {
      if(d==3)
      {
        if(c==7)
        {
           I[2]=jet(g,4)+var(1)^3*var(2)^2;
           return(I);
        }
        if(c==8)
        {
           I[2]=jet(g,4)+var(1)^4*var(2)^2;
           return(I);
        }
        if(c==9)
        {
           I[2]=jet(g,4);
           return(I);
        }
      }
      if((c==d+3)&&(d>=4))
      {
         return(ideal(var(1),var(2)^4+var(1)^2*var(2)^2+var(1)^d*var(2)));
      }
   }
   if((mu==5)&&(c==7)&&(d==3))
   {
      return(ideal(var(1),var(2)^4+var(1)^3*var(2)-3/2*var(1)^2*var(2)^2+var(1)^3*var(2)^2));
   }
   if((mu==6)&&(c==8)&&(d==3))
   {
      return(ideal(var(1),var(2)^4+var(1)^3*var(2)-3/2*var(1)^2*var(2)^2+var(1)^4*var(2)^2));
   }
   if((mu==c-2)&&(c>=9)&&(d==3))
   {
      I=ideal(var(1),var(2)^4+var(1)^3*var(2)-3/2*var(1)^2*var(2)^2+var(1)^(c-3)*var(2));
      return(I);
   }
   if((mu==7)&&(c==d+4)&&(d>=5))
   {
      return(ideal(var(1),var(2)^4+var(1)^3*var(2)^2+var(1)^d*var(2)));
   }
   if((mu==2*d-2)&&(2*d<=c)&&(c<=3*d)&&((d==4)||(d==5)))
   {
      return(ideal(var(1),var(2)^4+var(1)^d*var(2)+var(1)^(c-d-1)*var(2)^2));
   }
   return(ideal(0,0));
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),(c,ds);
   poly f1=x;
   poly f2=xy+y6+y9;
   poly f11=f1+f2*f1;
   poly f22=f2+f1^2;
   map phi=basering,x+y,y+y2;
   f1=phi(f11);
   f2=phi(f22);
   ideal I=f1,f2;
   classifyUnimodalMaps(I);
}
///////////////////////////////////////////////////////////////////////////////
static proc modulus(poly g)
// g=xy+y6+h.o.t. or xy2+y6+h.o.t.
{
   number a,b,c;
   matrix m=coef(g,var(2));
   if(m[1,1]==var(2)^9)
   {
      c=leadcoef(m[2,1]);
      if(m[1,2]==var(2)^8)
      {
         b=leadcoef(m[2,2]);
         if(m[1,3]==var(2)^7)
         {
            a=leadcoef(m[2,3]);
         }
      }
      else
      {
         if(m[1,2]==var(2)^7)
         {
           a=leadcoef(m[2,2]);
         }
      }
   }
   else
   {
      if(m[1,1]==var(2)^8)
      {
         b=leadcoef(m[2,1]);
         if(m[1,2]==var(2)^7)
         {
            a=leadcoef(m[2,2]);
         }
      }
      else
      {
         if(m[1,1]==var(2)^7)
         {
           a=leadcoef(m[2,1]);
         }
      }
   }
   if(leadmonom(g)==var(1)*var(2))
   {
      if(a!=0)
      {
         g=var(1)*var(2)+var(2)^6+(5*b-3*a^2)/(5*a^2)*var(2)^8+(25*c+14*a^3-35*a*b)/(25*a^3)*var(2)^9;
      }
   }
   if(leadmonom(g)==var(1)*var(2)^2)
   {
      if(a!=0)
      {
         g=var(1)*var(2)^2+var(2)^6+var(2)^7+(4*c*a-5*b)/(4*a^2)*var(2)^9;
      }
   }
   return(g);
}
///////////////////////////////////////////////////////////////////////////////
static proc doubleFoldNumber(poly g)
"USAGE:   doubleFoldNumber(g) , g poly of 2 variables
RETURN:   the double fold number of (x,g(x,y))
EXAMPLE:  example doubleFoldNumber; shows an example"
{
   if(nvars(basering)!=2){ERROR("designed for 2 variables");}
   def R=basering;
   ring S=0,(x,y,t),ds;
   map phi=R,x,y;
   poly g=phi(g);
   poly h=subst(g,y,y+t);
   poly f=diff(g,y);
   h=(h-g-t*f)/t2;
   ideal I=f,h,diff(h,t);
   int d=vdim(std(I)) div 2;
   setring R;
   return(d);
}
example
{
  "EXAMPLE"; echo=2;
   ring R=0,(x,y),ds;
   poly g=xy+y5+y7;
   doubleFoldNumber(g);
}
///////////////////////////////////////////////////////////////////////////////
/*
============================== examples of Rieger  ==============

ring R=0,(x,y),(c,ds);
poly f1=x;

poly f2=xy+y3;     //2
poly f2=y3+x2y;    //3
poly f2=xy+y4;     //3
poly f2=xy+y5+y7;  //4
poly f2=xy+y5;     //5
poly f2=xy2+y4+y5; //4
poly f2=xy2+y5+y6; //5
poly f2=xy2+y5+y9; //6
poly f2=xy2+y5;    //7
poly f2=x2y+y4+y5; //5
poly f2=x2y+y4;    //6

poly f11=f1+f2*f1;
poly f22=f2+f1^2;
map phi=basering,x+y,y+y2;
f1=phi(f11);
f2=phi(f22);
ideal I=f1,f2;
coDimMap(I,1);
classifySimpleMaps(I);

==================== Examples of Gibson and Hobbs =============

ring R=0,t,(c,ds);
poly f1=t4;
poly f2=t7+t9;
poly f3=t17;
ideal I=f1,f2,f3;
map phi=R,t+2t2+3t3+7t4;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
int k=12;
poly f1=t4;
poly f2=t6+t^(2*k-7);
poly f3=t^(2*k+1);
f1=f1+f2*f3;
f2=f2+f3^2;
f3=f3+f1*f2*f3;
ideal I=f1,f2,f3;
map phi=R,t+t2+2t3+5t4;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
int k=8;
poly f1=t4;
poly f2=t6+t^(2*k+1);
f1=f1+f2*f1;
f2=f2+f1^2;
ideal I=f1,f2,0;
map phi=R,t+t2+3t3+5t4;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
int k=8;
poly f1=t2;
poly f2=t^(2*k+1);
f1=f1+f2*f1;
f2=f2+f1^2;
ideal I=f1,f2,0;
map phi=R,t+t2+3t3+5t4;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
int k=8;
int p=11;
poly f1=t3;
poly f2=t^(3*k+1)+t^(3*p+2);
f1=f1+f2*f1;
f2=f2+f1^2;
ideal I=f1,f2,0;
map phi=R,t+t2+3t3+5t4;
I=phi(I);
coDimMap(I);

============================  more examples ======================

ring R=0,t,(c,ds);
poly f1=t8;
poly f2=t10+t13;
poly f3=t12+2t15;
f1=f1+231*f2*f1+311*f1*f3+71*f1^2+611*f1*f2*f3;
f2=f2+911*f1^2+511*f2*f3+f1^2*f2^2;
f3=f3+731*f1*f2*f3+1171*f3^2+f2^3;
ideal I=f1,f2,f3;
map phi=R,t+t2+3t3+5t4+7t6;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
poly f1=t12;
poly f2=t20+t30+t35;
f1=f1+231*f2*f1+71*f1^2;
f2=f2+911*f1^2+f1^2*f2^2;
ideal I=f1,f2;
map phi=R,t+t2+3t3+5t4+7t6;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
poly f1=t16;
poly f2=t24+t28+t30+t35;
f1=f1+231*f2*f1+71*f1^2;
f2=f2+911*f1^2+f1^2*f2^2;
ideal I=f1,f2;
map phi=R,t+t2+3t3+5t4+7t6;
I=phi(I);
coDimMap(I);

ring R=0,t,(c,ds);
int k=7;
poly f1=t^(8*k);
poly f2=t^(10*k)+t^(13*k);
poly f3=t^(12*k)+2t^(15*k);
poly f4=t101;
f1=f1+231*f2*f1+311*f1*f3+71*f1^2+611*f1*f2*f3;
f2=f2+911*f1^2+511*f2*f3+f1^2*f2^2;
f3=f3+731*f1*f2*f3+1171*f3^2+f2^3;
ideal I=f1,f2,f3,f4;
map phi=R,t+22t2+323t3+555t4+777t6;
I=phi(I);
module M=jacob(I);
module N=freemodule(ncols(I));
list Re=modVStd(M,N,I,250);
Re=modVStd0(M,N,I,250);

ring R=0,(x,y),(c,ds);
ideal I=x,xy2+y5+y9;
map phi=R,x+xy4,y+x2+y11;
I=phi(I);
module M=maxideal(1)*jacob(I);
module N=I*freemodule(2);
list Re=modVStd(M,N,I,15);

 ring R=0,(x,y),(c,ds);
 poly f1=x;
 poly f2=xy+y5+y7;
 poly f11=f1+f2*f1;
 poly f22=f2+f1^2;
 map phi=basering,x+y,y+y2;
 f1=phi(f11);
 f2=phi(f22);
 ideal I=f1,f2;
 map si=R,x+xy3+y11,y+x3+y14+xy17;
 I=si(I);
 module M=maxideal(1)*jacob(I);
 module N=I*freemodule(2);
 list Re=modVStd(M,N,I,15);

 ring R=0,(x,y),(c,ds);
 ideal I=x,x2y+y4;
 map phi=R,x+xy4,y+x2+y11;
 I=phi(I);
 module M=maxideal(1)*jacob(I);
 module N=I*freemodule(2);
 list Re=modVStd(M,N,I,15);

ring R=0,t,(c,ds);
int k=5;
poly f1=t^(16*k);
poly f2=t^(24*k)+t^(28*k)+t^(30*k)+t^(35*k);
poly f3=t181;
f1=f1+231*f2*f1+71*f1^2;
f2=f2+911*f1^2+f1^2*f2^2;
f3=f3+f1*f2;
ideal I=f1,f2,f3;
map phi=R,t+55t2+366t3+577t4+788t6;
I=phi(I);
module M=jacob(I);
module N=freemodule(ncols(I));
list Re=modVStd(M,N,I,300);

ring R=0,t,(c,ds);
int k=7;
poly f1=t^(12*k);
poly f2=t^(20*k)+t^(30*k)+t^(35*k);
poly f3=t139;
f1=f1+231*f2*f1+715*f1^2;
f2=f2+911*f1^2+4567*f1^2*f2^2;
f3=f3+333*f1*f2;
ideal I=f1,f2;
map phi=R,t+22t2+333t3+544t4+755t6+567t7;
I=phi(I);
module M=jacob(I);
module N=freemodule(ncols(I));
list Re=modVStd(M,N,I,350);
*/