source: git/Singular/LIB/noether.lib @ 431fe1

// AH/GP last modified:  12.05.2006
//////////////////////////////////////////////////////////////////////////////
version = "$Id: noether.lib,v 1.3 2006-12-11 18:57:42 Singular Exp $";
category="Commutative Algebra";
info="
LIBRARY: noether.lib   Noether normalization of an ideal (not nessecary homogeneous)
AUTHORS: A. Hashemi,  Amir.Hashemi@lip6.fr


OVERVIEW:
 A library for computing the Noether normalization of an ideal that DOES NOT require the computation of the dimension of the ideal.
It checks whether an ideal is in Noether position.  A modular version of these algorithms is also provided.
The procedures are based on a paper of Amir Hashemi 'Efficient Algorithms for Computing Noether Normalization'
Submitted to: Special Issue of Mathematics in Computer Science on Symbolic and Numeric Computation.

This library computes also Castelnuovo-Mumford regularity and satiety of an ideal.  A modular version of these algorithms is also provided.
The procedures are based on a paper of Amir Hashemi 'Computation of  Castelnuovo-Mumford regularity and satiety'
Submitted to: ISSAC 2007.


PROCEDURES:
 NP_test(id);  checks whether monomial ideal id is in Noether position
 MNP_test(id);  checks whether ideal id is in Noether position by modular methods
 NP(id);  Noether normalization of ideal id
 MNP(id);  Noether normalization of ideal id by modular methods
 nsatiety(id);  Satiety of ideal id
 msatiety(id)  Satiety of ideal id by modular methods
 regCM(id);  Castelnuovo-Mumford regularity of ideal id
 mregCM(id);  Castelnuovo-Mumford regularity of ideal id by modular methods
";
LIB "elim.lib";
LIB "algebra.lib";
LIB "poly.lib";
LIB "ring.lib";
LIB "presolve.lib";

///////////////////////////////////////////////////////////////////////////////
proc NP_test (ideal I)
"
USAGE:  NP-test (I); I monomial ideal
RETURN:  A list which first element is 1 if i is in Noether position
  0  otherwise. The second element of this list is the list of variable which
  its first part is the variable such that a power of this varaibles belong to the initial of i.
  It return also the dimension of i if i is in Noether position
ASSUME:  i is a nonzero monomial ideal.
"
{
//--------------------------- initialisation ---------------------------------
   int  time,ii,j,k,l,d,t,jj;
   intvec v;
   def r0 = basering;
   int n = nvars(r0)-1;
   list L,Y,P1,P2,P3;
   if (I[1]==1)
   {
   print("The ideal is 1");return(1);
   }
   for ( ii = 1; ii <= n+1; ii++ )
     {
  L[ii]=0;
     }
   for ( ii = 1; ii <= size(I); ii++ )
   {
    Y=findvars(I[ii],1)[1];
  l=rvar(Y[1][1]);
  if (size(Y[1])==1)
  {
    L[l]=1;
    P1=insert(P1,Y[1][1]);
  }
  if (L[l]==0)
  {
    L[l]=-1;
  }
    }
   t=size(P1);
   if (t==0)
   {
         for ( jj = 1; jj <= n+1; jj++ )
         {
         P3=insert(P3,varstr(jj));
         }
   }
   else
   {
         P2=findvars(ideal(P1[1..t]),1)[3];
         for ( jj = 1; jj <= size(P2[1]); jj++ )
         {
         P3=insert(P3,P2[1][jj]);
         }
   }
   if (L[n+1]==-1)
        {
     return(list(0,P1+P3));
  }
   for ( ii = 1; ii <= n; ii++ )
     {
  if (L[ii]==-1)
        {
     return(list(0,P1+P3));
  }
  if (L[ii]==0 and L[ii+1]==1)
        {
     return(list(0,P1+P3));
  }
    }
    d=n+1-sum(L);
    print("The dimension of the ideal is:");print(d);
    return(list(1,P1+P3));

}
//////////////////////////////////////////
proc MNP_test (ideal i)
"USAGE:  MNP-test (i); i an ideal
RETURN:  1 if i is in Noether position 0  otherwise.
NOTE:  This test is a probabilistic test, and it computes the initial of the ideal modulo the prime number 2147483647 (the biggest prime less than 2^31).
"
{
       "// WARNING:
// The procedure is probabilistic and  it computes the initial of the ideal modulo the prime number 2147483647";
   int p;
   def br=basering;
   setring br;
   ideal I;
   list #;
   option(redSB);
   p=2147483647;
   #=ringlist(br);
   #[1]=p;
   def oro=ring(#);
   setring oro;
   ideal sbi,lsbi;
   sbi=fetch(br,i);
   lsbi=lead(std(sbi));
   setring br;
   I=fetch(oro,lsbi);
   I=simplify(I,1);
   attrib(I,"isSB",1);
   return(NP_test(I));
}


///////////////////////////////////////////////////////////////////////////////
proc NP (ideal i)
"
USAGE:   NP (i); i ideal
RETURN:  A linear map phi such that  phi(i) is in Noether position
"
{
//--------------------------- initialisation ---------------------------------
   int ii,jj,d,time,n,nl;
   intmat ran;
   def r0 = basering;
   ideal K,chcoord;
   n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,K,chcoord,m,L;
   list #;
   poly P;
   map phi;
   i = fetch(r0,i);
   time=rtimer;
   system("--ticks-per-sec",10);
   i=std(i);
   sbi=sort(lead(i))[1];
   #=NP_test(sbi);
   if ( #[1]== 1 )
     {
  return ("The ideal is in Noether position and the time of this computation is:",rtimer-time,"/10 sec.");
     }
   else
     {
    L=maxideal(1);
    chcoord=maxideal(1);
     for ( ii = 1; ii<=n+1; ii++ )
    {
       chcoord[rvar(#[2][ii])]=L[ii];
    }
    phi=r1,chcoord;
                sbi=phi(sbi);
                if ( NP_test(sbi)[1] == 1 )
              {
            setring r0;
            chcoord=fetch(r1,chcoord);
     return (chcoord,"and the time of this computation is:",rtimer-time,"/10 sec.");
                }
     }
   while ( nl < 30 )
   {
     nl=nl+1;
     I=i;
     L=maxideal(1);
     for ( ii = n; ii>=0; ii-- )
     {
       chcoord=select1(maxideal(1),1,(ii));
       ran=random(100,1,ii);
       ran=intmat(ran,1,ii+1);
       ran[1,ii+1]=1;
       m=select1(maxideal(1),1,(ii+1));
       for ( jj = 1; jj<=ii+1; jj++ )
       {
           P=P+ran[1,jj]*m[jj];
       }
       chcoord[ii+1]=P;
  L[ii+1]=P;
       P=0;
       phi=r1,chcoord;
       I=phi(I);
       if ( NP_test(sort(lead(std(I)))[1])[1] == 1 )
         {
     K=x(ii..n);
            setring r0;
            K=fetch(r1,K);
                 ideal L=fetch(r1,L);
     return (L,"and the time of this computation is:",rtimer-time,"/10 sec.");
           }

     }
   }
       "// WARNING:
// The procedure has entered in more than 30 loops: in your example
// the method may enter an infinite loop over a finite field!";
       return (-1);
}
///////////////////////////////////////////////////////////////////////////////
proc MNP (ideal i)
"USAGE:  MNP (i); i ideal
RETURN:  A linear map phi such that  phi(i) is in Noether position
NOTE:  It uses the procedure  MNP_test to test Noether position.
"
{
//--------------------------- initialisation ---------------------------------
   int ii,jj,d,time,n,nl;
   intmat ran;
   def r0 = basering;
   ideal K,chcoord;
   n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,K,chcoord,m,L;
   poly P;
   list #;
   map phi;
   i = fetch(r0,i);
   time=rtimer;
   system("--ticks-per-sec",10);
    #=MNP_test(i);
   if ( #[1]== 1 )
     {
  return ("The ideal is in Noether position and the time of this computation is:",rtimer-time,"/10 sec.");
     }
   else
     {
    L=maxideal(1);
    chcoord=maxideal(1);
     for ( ii = 1; ii<=n+1; ii++ )
    {
       chcoord[rvar(#[2][ii])]=L[ii];
    }
    phi=r1,chcoord;
                I=phi(i);
                if ( MNP_test(I)[1] == 1 )
              {
            setring r0;
            chcoord=fetch(r1,chcoord);
     return (chcoord,"and the time of this computation is:",rtimer-time,"/10 sec.");
                }
     }
   while ( nl < 30 )
   {
     nl=nl+1;
     I=i;
     L=maxideal(1);
     for ( ii = n; ii>=0; ii-- )
     {
       chcoord=select1(maxideal(1),1,(ii));
       ran=random(100,1,ii);
       ran=intmat(ran,1,ii+1);
       ran[1,ii+1]=1;
       m=select1(maxideal(1),1,(ii+1));
       for ( jj = 1; jj<=ii+1; jj++ )
       {
           P=P+ran[1,jj]*m[jj];
       }
       chcoord[ii+1]=P;
       L[ii+1]=P;
       P=0;
       phi=r1,chcoord;
       I=phi(I);
       if ( MNP_test(I)[1] == 1 )
         {
     K=x(ii..n);
            setring r0;
            K=fetch(r1,K);
                 ideal L=fetch(r1,L);
     return (L,"and the time of this computation is:",rtimer-time,"/10 sec.");
           }

     }
   }
       "// WARNING:
// The procedure has entered in more than 30 loops: in your example
// the method may enter an infinite loop over a finite field!";
       return (-1);
}

////////////////////////////////////////////////////////////////////////////////////
proc Test (ideal i)
"
USAGE:   Test (i); i a monomial ideal,
RETURN:  1 if the last variable is in generic position for i and 0 otherwise.
THEORY:  The last variable is in generic position if the quotient of the ideal
   with respect to this variable is equal to the quotient of the ideal with respect to the maximal ideal.
"
{
//--------------------------- initialisation ---------------------------------
   int n,ret;
   def r0 = basering;
   n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal I,i;
   i = fetch(r0,i);
   attrib(i,"isSB",1);
   I=quotient(select(i,n+1),x(n));
   I=I*maxideal(1);
   ret=1;
   if (size(reduce(I,i)) <> 0)
   {
  ret=0;
   }
return(ret);
}


////////////////////////////////////////////////////////////////////////////////////
proc nsatiety (ideal i)
"
USAGE:   nsatiety (i); i ideal,
RETURN:  an integer, the satiety of i.
         (returns -1 if i is not homogeneous)
ASSUME:  i is a homogeneous ideal of the basering R=K[x(0)..x(n)].
THEORY:  The satiety, or saturation index, of a homogeneous ideal i is the
         least integer s such that, for all d>=s, the degree d part of the
         ideals i and isatiety=satiety(i,maxideal(1))[1] coincide.
"
{
//--------------------------- initialisation ---------------------------------
   int e,ii,jj,h,d,time,lastv,nl,ret;
   intmat ran;
   def r0 = basering;
   int n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,K,chcoord,m,L;
   poly P;
   map phi;
   i = fetch(r0,i);
   time=rtimer;
   system("--ticks-per-sec",100);
   sbi=std(i);
//----- Check ideal homogeneous
   if ( homog(sbi) == 0 )
   {
       dbprint(2,"The ideal is not homogeneous, and time for this test is: " + string(rtimer-time) + "/100sec.");
  return ();
   }
   I=simplify(lead(sbi),1);
   attrib(I,"isSB",1);
   K=select(I,n+1);
   if (size(K) == 0)
   {
  dbprint(2,"satiety(i)=0 and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
   }
   if (Test(I) == 1 )
   {
  dbprint(2,"satiety(i)=" + string(maxdeg1(K)) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
   }
   while ( nl < 5 )
   {
     nl=nl+1;
     chcoord=select1(maxideal(1),1,(n));
     ran=random(100,1,n);
     ran=intmat(ran,1,n+1);
     ran[1,n+1]=1;
     m=select1(maxideal(1),1,(n+1));
     for ( jj = 1; jj<=n+1; jj++ )
       {
           P=P+ran[1,jj]*m[jj];
       }
     chcoord[n+1]=P;
     P=0;
     phi=r1,chcoord;
     L=std(phi(i));
     I=simplify(lead(L),1);
     attrib(I,"isSB",1);
     K=select(I,n+1);
     if (size(K) == 0)
     {
  dbprint(2,"satiety(i)=0 and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
     }
     if (Test(I) == 1 )
     {
  dbprint(2,"satiety(i)=" + string(maxdeg1(K)) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
     }
   }
}


//////////////////////////////////////////////////////////////////////////////
proc msatiety (ideal i)
"
USAGE:   msatiety (i); i ideal,
RETURN:  an integer, the satiety of i.
         (returns -1 if i is not homogeneous)
ASSUME:  i is a homogeneous ideal of the basering R=K[x(0)..x(n)].
THEORY:  The satiety, or saturation index, of a homogeneous ideal i is the
         least integer s such that, for all d>=s, the degree d part of the
         ideals i and isatiety=satiety(i,maxideal(1))[1] coincide.
NOTE:   This is a probabilistic procedure, and it computes the initial of the ideal modulo the prime number 2147483647 (the biggest prime less than 2^31).
"
{
//--------------------------- initialisation ---------------------------------
       "// WARNING:
// The procedure is probabilistic and  it computes the initial of the ideal modulo the prime number 2147483647";
   int e,ii,jj,h,d,time,lastv,nl,ret,s1,d1,siz,j,si,u,k,p;
   intvec v1;
   intmat ran;
   def r0 = basering;
   int n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,K,chcoord,m,L,sbi1,lsbi1,id1;
   vector V1;
   list #,LL,PL,Gb1,VGb1,Gb2,VGb2,Res1,Res2;
   poly P;
   map phi;
   time=rtimer;
   system("--ticks-per-sec",100);
   i = fetch(r0,i);
//----- Check ideal homogeneous
   if ( homog(i) == 0 )
   {
       "// WARNING: The ideal is not homogeneous.";
       dbprint(2,"Time for this test is: " + string(rtimer-time) + "/100sec.");
  return ();
   }
   option(redSB);
   p=2147483647;
   list r2=ringlist(r1);
   r2[1]=p;
   def oro=ring(r2);
   setring oro;
   ideal sbi=fetch(r1,i);
   sbi=std(sbi);
   setring r1;
   sbi=fetch(oro,sbi);
   kill oro;
   I=simplify(lead(sbi),1);
   attrib(I,"isSB",1);
   K=select(I,n+1);
   if (size(K) == 0)
   {
  dbprint(2,"msatiety(i)=0 and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
   }
   if (Test(I) == 1 )
   {
  dbprint(2,"msatiety(i)=" + string(maxdeg1(K)) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
   }
   while ( nl < 30 )
   {
     nl=nl+1;
     chcoord=select1(maxideal(1),1,(n));
     ran=random(100,1,n);
     ran=intmat(ran,1,n+1);
     ran[1,n+1]=1;
     m=select1(maxideal(1),1,(n+1));
     for ( jj = 1; jj<=n+1; jj++ )
       {
           P=P+ran[1,jj]*m[jj];
       }
     chcoord[n+1]=P;
     P=0;
     phi=r1,chcoord;
     sbi=phi(i);
     list r2=ringlist(r1);
     r2[1]=p;
     def oro=ring(r2);
     setring oro;
     ideal sbi=fetch(r1,sbi);
     sbi=std(sbi);
     setring r1;
     sbi=fetch(oro,sbi);
     kill oro;
     lsbi1=lead(sbi);
     attrib(lsbi1,"isSB",1);
     K=select(lsbi1,n+1);
     if (size(K) == 0)
     {
  dbprint(2,"msatiety(i)=0 and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
     }
     if (Test(lsbi1) == 1 )
     {
  dbprint(2,"msatiety(i)=" + string(maxdeg1(K)) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
  return();
     }
   }
}

//////////////////////////////////////////////////////////////////////////////
//
proc regCM (ideal i)
"
USAGE:  regCM (i); i ideal
RETURN:  the Castelnuovo-Mumford regularity of i.
         (returns -1 if i is not homogeneous)
ASSUME:  i is a homogeneous ideal.
"
{
//--------------------------- initialisation ---------------------------------
   int e,ii,jj,H,h,d,time,lastv,nv,nl;
   int lastind,ch,nesttest,NPtest,nl,N,acc;
   intmat ran;
   def r0 = basering;
   int n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,J,K,chcoord,m,L;
   list #;
   poly P;
   map phi;
   i = fetch(r0,i);
   time=rtimer;
   sbi=std(i);
//----- Check ideal homogeneous
   if ( homog(sbi) == 0 )
   {
       "// The ideal is not homogeneous!";
       return (-1);
   }
   I=simplify(lead(sbi),1);
   attrib(I,"isSB",1);
   d=dim(I);
   K=select(I,n+1);
   if (size(K) == 0)
   {
  h=0;
   }
   if (Test(I) == 1)
   {
  h=maxdeg1(K);
   }
   else
   {
  while ( nl < 5 )
  {
    nl=nl+1;
         chcoord=select1(maxideal(1),1,(n));
         ran=random(100,1,n);
         ran=intmat(ran,1,n+1);
         ran[1,n+1]=1;
         m=select1(maxideal(1),1,(n+1));
         for ( jj = 1; jj<=n+1; jj++ )
           {
             P=P+ran[1,jj]*m[jj];
           }
         chcoord[n+1]=P;
         P=0;
         phi=r1,chcoord;
    i=phi(i);
         I=simplify(lead(std(i)),1);
         attrib(I,"isSB",1);
         K=select(I,n+1);
         if (size(K) == 0)
         {
      h=0;break
         }
         if (Test(I) == 1 )
         {
      h=maxdeg1(K);break;
    }
  }
    }
   for ( ii = n; ii>=n-d+1; ii-- )
   {
  i=subst(i,x(ii),0);
  s = "ring mr = ",charstr(r0),",x(0..ii-1),dp;";
  execute(s);
  ideal i,sbi,I,J,K,chcoord,m,L;
     poly P;
     map phi;
  i=imap(r1,i);
  sbi=std(i);
        I=simplify(lead(sbi),1);
  attrib(I,"isSB",1);
       K=select(I,ii);
       if (size(K) == 0)
       {
    H=0;
       }
  if (Test(I) == 1)
  {
    H=maxdeg1(K);
     }
     else
     {
    while ( nl < 5 )
    {
      nl=nl+1;
           chcoord=select1(maxideal(1),1,(ii-1));
           ran=random(100,1,ii-1);
           ran=intmat(ran,1,ii);
           ran[1,ii]=1;
           m=select1(maxideal(1),1,(ii));
           for ( jj = 1; jj<=ii; jj++ )
             {
               P=P+ran[1,jj]*m[jj];
             }
           chcoord[ii]=P;
           P=0;
           phi=mr,chcoord;
           i=phi(i);
           I=simplify(lead(std(i)),1);
           attrib(I,"isSB",1);
           K=select(I,ii);
           if (size(K) == 0)
           {
        H=0;break;
           }
           if (Test(I) == 1 )
           {
        H=maxdeg1(K);break;
      }
    }
    setring r1;
    i=imap(mr,i);
    kill mr;
      }
  if (H > h)
  {
    h=H;
  }
   }
dbprint(2,"regCM(i)=" + string(h) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
return();
}
//////////////////////////////////////////////////////////////////////////////
//
proc mregCM (ideal i)
"
USAGE:  mregCM (i); i ideal
RETURN:  an integer, the Castelnuovo-Mumford regularity of i.
         (returns -1 if i is not homogeneous)
ASSUME:  i is a homogeneous ideal.
NOTE:   This is a probabilistic procedure, and it computes the initial of the ideal modulo the prime number 2147483647 (the biggest prime less than 2^31).
"
{
//--------------------------- initialisation ---------------------------------
       "// WARNING:
// The procedure is probabilistic and  it computes the initial of the ideal modulo the prime number 2147483647";
   int e,ii,jj,H,h,d,time,lastv,sat,firstind,nv,nl,p;
   int lastind,ch,nesttest,NPtest,nl,N,acc;
   intmat ran;
   def r0 = basering;
   int n = nvars(r0)-1;
   string s = "ring r1 = ",charstr(r0),",x(0..n),dp;";
   execute(s);
   ideal i,sbi,I,J,K,chcoord,m,L,lsbi1,lsbi2;
   list #;
   poly P;
   map phi;
   i = fetch(r0,i);
   time=rtimer;
//----- Check ideal homogeneous
   if ( homog(i) == 0 )
   {
       "// The ideal is not homogeneous!";
       return (-1);
   }
   option(redSB);
   p=2147483647;
   #=ringlist(r1);
   #[1]=p;
   def oro=ring(#);
   setring oro;
   ideal sbi,lsbi;
   sbi=fetch(r1,i);
   lsbi=lead(std(sbi));
   setring r1;
   lsbi1=fetch(oro,lsbi);
   lsbi1=simplify(lsbi1,1);
   attrib(lsbi1,"isSB",1);
   kill oro;
   I=lsbi1;
   d=dim(I);
   K=select(I,n+1);
   if (size(K) == 0)
   {
  h=0;
   }
   if (Test(I) == 1)
   {
  h=maxdeg1(K);
   }
   else
   {
  while ( nl < 5 )
  {
    nl=nl+1;
         chcoord=select1(maxideal(1),1,(n));
         ran=random(100,1,n);
         ran=intmat(ran,1,n+1);
         ran[1,n+1]=1;
         m=select1(maxideal(1),1,(n+1));
         for ( jj = 1; jj<=n+1; jj++ )
           {
             P=P+ran[1,jj]*m[jj];
           }
         chcoord[n+1]=P;
         P=0;
         phi=r1,chcoord;
    i=phi(i);
       #=ringlist(r1);
       #[1]=p;
       def oro=ring(#);
       setring oro;
       ideal sbi,lsbi;
       sbi=fetch(r1,i);
       lsbi=lead(std(sbi));
       setring r1;
       lsbi1=fetch(oro,lsbi);
       lsbi1=simplify(lsbi1,1);
       attrib(lsbi1,"isSB",1);
       kill oro;
    I=lsbi1;
         K=select(I,n+1);
         if (size(K) == 0)
         {
      h=0;break
         }
         if (Test(I) == 1 )
         {
      h=maxdeg1(K);break;
    }
  }
    }
   for ( ii = n; ii>=n-d+1; ii-- )
   {
  i=subst(i,x(ii),0);
  s = "ring mr = ",charstr(r0),",x(0..ii-1),dp;";
  execute(s);
  ideal i,sbi,I,J,K,chcoord,m,L,lsbi1;
     poly P;
        list #;
     map phi;
  i=imap(r1,i);
     #=ringlist(mr);
     #[1]=p;
     def oro=ring(#);
     setring oro;
     ideal sbi,lsbi;
     sbi=fetch(mr,i);
     lsbi=lead(std(sbi));
     setring mr;
     lsbi1=fetch(oro,lsbi);
     lsbi1=simplify(lsbi1,1);
     attrib(lsbi1,"isSB",1);
     kill oro;
  I=lsbi1;
       K=select(I,ii);
       if (size(K) == 0)
       {
    H=0;
       }
  if (Test(I) == 1)
  {
    H=maxdeg1(K);
     }
     else
     {
    while ( nl < 5 )
    {
      nl=nl+1;
           chcoord=select1(maxideal(1),1,(ii-1));
           ran=random(100,1,ii-1);
           ran=intmat(ran,1,ii);
           ran[1,ii]=1;
           m=select1(maxideal(1),1,(ii));
           for ( jj = 1; jj<=ii; jj++ )
             {
               P=P+ran[1,jj]*m[jj];
             }
           chcoord[ii]=P;
           P=0;
           phi=mr,chcoord;
           i=phi(i);
         #=ringlist(mr);
         #[1]=p;
         def oro=ring(#);
         setring oro;
         ideal sbi,lsbi;
         sbi=fetch(mr,i);
         lsbi=lead(std(sbi));
         setring mr;
         lsbi1=fetch(oro,lsbi);
         lsbi1=simplify(lsbi1,1);
         attrib(lsbi1,"isSB",1);
         kill oro;
      I=lsbi1;
           K=select(I,ii);
           if (size(K) == 0)
           {
        H=0;break;
           }
           if (Test(I) == 1 )
           {
        H=maxdeg1(K);break;
      }
    }
    setring r1;
    i=imap(mr,i);
    kill mr;
      }
  if (H > h)
  {
    h=H;
  }
  }
dbprint(2,"mregCM(i)=" + string(h) + " and the time of this computation: " + string(rtimer-time) + "/100sec.");
return();
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(X,Y,a,b),dp;
poly f=X^8+a*Y^4-Y;
poly g=Y^8+b*X^4-X;
poly h=diff(f,X)*diff(g,Y)-diff(f,Y)*diff(g,X);
ideal i=f,g,h;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,a,b),dp;
ideal i=2*y^2*(y^2+x^2)+(b^2-3*a^2)*y^2-2*b*y^2*(x+y)+2*a^2*b*(y+x)-a^2*x^2+a^2*(a^2-b^2),4*y^3+4*y*(y^2+x^2)-2*b*y^2-4*b*y*(y+x)+2*(b^2-3*a^2)*y+2*a^2*b,4*x*y^2-2*b*y^2-2*a^2*x+2*a^2*b;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(t,a,b,c,d),dp;
ideal i=b4-a3d, ab3-a3c, bc4-ac3d-bcd3+ad4, c6-bc3d2-c3d3+bd5, ac5-b2c3d-ac2d3+b2d4, a2c4-a3d3+b3d3-a2cd3, b3c3-a3d3, ab2c3-a3cd2+b3cd2-ab2d3, a2bc3-a3c2d+b3c2d-a2bd3, a3c3-a3bd2, a4c2-a3b2d;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,e),dp;
ideal i=6*b4*c3+21*b4*c2*d+15b4cd2+9b4d3-8b2c2e-28b2cde+36b2d2e-144b2c-648b2d-120, 9b4c4+30b4c3d+39b4c2d2+18b4cd3-24b2c3e-16b2c2de+16b2cd2e+24b2d3e-432b2c2-720b2cd-432b2d2+16c2e2-32cde2+16d2e2+576ce-576de-240c+5184,-15b2c3e+15b2c2de-81b2c2+216b2cd-162b2d2+40c2e2-80cde2+40d2e2+1008ce-1008de+5184, -4b2c2+4b2cd-3b2d2+22ce-22de+261;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(c,b,d,p,q),dp;
ideal i=2*(b-1)^2+2*(q-p*q+p^2)+c^2*(q-1)^2-2*b*q+2*c*d*(1-q)*(q-p)+2*b*p*q*d*(d-c)+b^2*d^2*(1-2*p)+2*b*d^2*(p-q)+2*b*d*c*(p-1)+2*b*p*q*(c+1)+(b^2-2*b)*p^2*d^2+2*b^2*p^2+4*b*(1-b)*p+d^2*(p-q)^2,d*(2*p+1)*(q-p)+c*(p+2)*(1-q)+b*(b-2)*d+b*(1-2*b)*p*d+b*c*(q+p-p*q-1)+b*(b+1)*p^2*d, -b^2*(p-1)^2+2*p*(p-q)-2*(q-1),b^2+4*(p-q*q)+3*c^2*(q-1)*(q-1)-3*d^2*(p-q)^2+3*b^2*d^2*(p-1)^2+b^2*p*(p-2)+6*b*d*c*(p+q+q*p-1);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,e,f),dp;
ideal i=2adef+3be2f-cef2,4ad2f+5bdef+cdf2,2abdf+3b2ef-bcf2,4a2df+5abef+acf2,4ad2e+3bde2+7cdef, 2acde+3bce2-c2ef, 4abde+3b2e2-4acdf+2bcef-c2f2, 4a2de+3abe2+7acef, 4acd2+5bcde+c2df, 4abd2+3b2de+7bcdf, 16a2d2-9b2e2+32acdf-18bcef+7c2f2, 2abcd+3b2ce-bc2f, 4a2cd+5abce+ac2f, 4a2bd+3ab2e+7abcf, abc2f-cdef2, ab2cf-bdef2, 2a2bcf+3be2f2-cef3, ab3f-3bdf3, 2a2b2f-4adf3+3bef3-cf4, a3bf+4aef3, 3ac3e-cde3, 3b2c2e-bc3f+2cd2ef, abc2e-cde2f, 6a2c2e-4ade3-3be4+ce3f, 3b3ce-b2c2f+2bd2ef, 2a2bce+3be3f-ce2f2, 3a3ce+4ae3f, 4bc3d+cd3e, 4ac3d-3bc3e-2cd2e2+c4f, 8b2c2d-4ad4-3bd3e-cd3f, 4b3cd+3bd3f, 4ab3d+3b4e-b3cf-6bd2f2, 4a4d+3a3be+a3cf-8ae2f2;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z,t,u,v,w),dp;
ideal i=2tw+2wy-wz,2uw2-10vw2+20w3-7tu+35tv-70tw, 6tw2+2w2y-2w2z-21t2-7ty+7tz, 2v3-4uvw-5v2w+6uw2+7vw2-15w3-42vy, 6tw+9wy+2vz-3wz-21x, 9uw3-45vw3+135w4+14tv2-70tuw+196tvw-602tw2-14v2z+28uwz+14vwz-28w2z+147ux-735vx+2205wx-294ty+98tz+294yz-98z2, 36tw3+6w3y-9w3z-168t2w-14v2x+28uwx+14vwx-28w2x-28twy+42twz+588tx+392xy-245xz, 2uvw-6v2w-uw2+13vw2-5w3-28tw+14wy, u2w-3uvw+5uw2-28tw+14wy, tuw+tvw-11tw2-2vwy+8w2y+uwz-3vwz+5w2z-21wx, 5tuw-17tvw+33tw2-7uwy+22vwy-39w2y-2uwz+6vwz-10w2z+63wx, 20t2w-12uwx+30vwx-15w2x-10twy-8twz+4wyz, 4t2w-6uwx+12vwx-6w2x+2twy-2wy2-2twz+wyz, 8twx+8wxy-4wxz;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,x,w,u,v),dp;
ideal i=a+b+c+d,u+v+w+x, 3ab+3ac+3bc+3ad+3bd+3cd+2,bu+cu+du+av+cv+dv+aw+bw+dw+ax+bx+cx,bcu+bdu+cdu+acv+adv+cdv+abw+adw+bdw+abx+acx+bcx,abc+abd+acd+bcd,bcdu+acdv+abdw+abcx;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(b,x,y,z,s,t,u,v,w),dp;
ideal i=su+bv, tu+bw,tv+sw,sx+by,tx+bz,ty+sz,vx+uy,wx+uz,wy+vz;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(t,a,b,c,d,e,f,g,h),dp;
ideal i=a+c+d-e-h,2df+2cg+2eh-2h2-h-1,3df2+3cg2-3eh2+3h3+3h2-e+4h, 6bdg-6eh2+6h3-3eh+6h2-e+4h, 4df3+4cg3+4eh3-4h4-6h3+4eh-10h2-h-1, 8bdfg+8eh3-8h4+4eh2-12h3+4eh-14h2-3h-1, 12bdg2+12eh3-12h4+12eh2-18h3+8eh-14h2-h-1, -24eh3+24h4-24eh2+36h3-8eh+26h2+7h+1;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,e,f,g,h,k,l),dp;
ideal i=f2h-1,ek2-1,g2l-1, 2ef2g2hk2+f2g2h2k2+2ef2g2k2l+2f2g2hk2l+f2g2k2l2+ck2, 2e2fg2hk2+2efg2h2k2+2e2fg2k2l+4efg2hk2l+2fg2h2k2l+2efg2k2l2+2fg2hk2l2+2bfh, 2e2f2ghk2+2ef2gh2k2+2e2f2gk2l+4ef2ghk2l+2f2gh2k2l+2ef2gk2l2+2f2ghk2l2+2dgl, e2f2g2k2+2ef2g2hk2+2ef2g2k2l+2f2g2hk2l+f2g2k2l2+bf2, 2e2f2g2hk+2ef2g2h2k+2e2f2g2kl+4ef2g2hkl+2f2g2h2kl+2ef2g2kl2+2f2g2hkl2+2cek, e2f2g2k2+2ef2g2hk2+f2g2h2k2+2ef2g2k2l+2f2g2hk2l+dg2, -e2f2g2hk2-ef2g2h2k2-e2f2g2k2l-2ef2g2hk2l-f2g2h2k2l-ef2g2k2l2-f2g2hk2l2+a2;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(b,c,d,e,f,g,h,j,k,l),dp;
ideal i=-k9+9k8l-36k7l2+84k6l3-126k5l4+126k4l5-84k3l6+36k2l7-9kl8+l9, -bk8+8bk7l+k8l-28bk6l2-8k7l2+56bk5l3+28k6l3-70bk4l4-56k5l4+56bk3l5+70k4l5-28bk2l6-56k3l6+8bkl7+28k2l7-bl8-8kl8+l9, ck7-7ck6l-k7l+21ck5l2+7k6l2-35ck4l3-21k5l3+35ck3l4+35k4l4-21ck2l5-35k3l5+7ckl6+21k2l6-cl7-7kl7+l8, -dk6+6dk5l+k6l-15dk4l2-6k5l2+20dk3l3+15k4l3-15dk2l4-20k3l4+6dkl5+15k2l5-dl6-6kl6+l7, ek5-5ek4l-k5l+10ek3l2+5k4l2-10ek2l3-10k3l3+5ekl4+10k2l4-el5-5kl5+l6, -fk4+4fk3l+k4l-6fk2l2-4k3l2+4fkl3+6k2l3-fl4-4kl4+l5, gk3-3gk2l-k3l+3gkl2+3k2l2-gl3-3kl3+l4, -hk2+2hkl+k2l-hl2-2kl2+l3, jk-jl-kl+l2;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,x(0..10),dp;
ideal i=x(1)*x(0),x(1)*x(2),x(2)*x(3),x(3)*x(4),x(4)*x(5),x(5)*x(6),x(6)*x(7),x(7)*x(8),x(8)*x(9),x(9)*x(10),x(10)*x(0);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,e,f,g,h,j,k,l,m,n,o,p,q,s),dp;
ideal i=ag,gj+am+np+q,bl,nq,bg+bk+al+lo+lp+b+c,ag+ak+jl+bm+bn+go+ko+gp+kp+lq+a+d+f+h+o+p,gj+jk+am+an+mo+no+mp+np+gq+kq+e+j+q+s-1,jm+jn+mq+nq,jn+mq+2nq,gj+am+2an+no+np+2gq+kq+q+s,2ag+ak+bn+go+gp+lq+a+d,bg+al, an+gq, 2jm+jn+mq, gj+jk+am+mo+2mp+np+e+2j+q, jl+bm+gp+kp+a+f+o+2p,lp+b,jn+mq,gp+a;
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(a,b,c,d,e,f,g,h,v,w,k,l,m,n,o,p,q,s,t,u),dp;
ideal i=af+bg+ch+dv+ew-1/2, a2f+b2g+c2h+d2v+e2w-1/3,tdw+agk+ahl+bhm+avn+bvo+cvp+awq+bwu+cws-1/6, a3f+b3g+c3h+d3v+e3w-1/4, tdew+abgk+achl+bchm+advn+bdvo+cdvp+aewq+bewu+cews-1/8, td2w+a2gk+a2hl+b2hm+a2vn+b2vo+c2vp+a2wq+b2wu+c2ws-1/12, ahkm+tawn+tbwo+avko+tcwp+avlp+bvmp+awku+awls+bwms-1/24, a4f+b4g+c4h+d4v+e4w-1/5, tde2w+ab2gk+ac2hl+bc2hm+ad2vn+bd2vo+cd2vp+ae2wq+be2wu+ce2ws-1/10, td2ew+a2bgk+a2chl+b2chm+a2dvn+b2dvo+c2dvp+a2ewq+b2ewu+c2ews-1/15,achkm+taewn+tbewo+advko+tcewp+advlp+bdvmp+aewku+aewls+bewms-1/30,t2d2w+a2gk2+a2hl2+2abhlm+b2hm2+a2vn2+2abvno+b2vo2+2acvnp+2bcvop+c2vp2+2tadwq+a2wq2+2tbdwu+2abwqu+b2wu2+2tcdws+2acwqs+2bcwus+c2ws2-1/20,td3w+a3gk+a3hl+b3hm+a3vn+b3vo+c3vp+a3wq+b3wu+c3ws-1/20,abhkm+tadwn+tbdwo+abvko+tcdwp+acvlp+bcvmp+abwku+acwls+bcwms-1/40,a2hkm+ta2wn+tb2wo+a2vko+tc2wp+a2vlp+b2vmp+a2wku+a2wls+b2wms-1/60,tawko+tawlp+tbwmp+avkmp+awkms-1/20;
}