//SINGULAR Procedure4.6.3


proc prepareQuotientring(ideal i)
"USAGE:   prepareQuotientring(i); i standard basis
RETURN:   a list l of two strings:
          l[1] to define K[x\u,u ], u a maximal independent 
          set for i
          l[2] to define K(u)[x\u ], u a maximal independent 
          set for i
          both rings with lexicographical ordering
EXAMPLE: example prepareQuotientring; shows an example
"  
{
  string va,pa;
//v describes the independent set u: var(j) is in 
//u iff v[j]!=0
  intvec v=indepSet(i); 
  int k;

  for(k=1;k<=size(v);k++)
  {
    if(v[k]!=0)
    {
      pa=pa+"var("+string(k)+"),";
    }
    else
    {
      va=va+"var("+string(k)+"),";
    }
  }
  
  pa=pa[1..size(pa)-1];
  va=va[1..size(va)-1];
  
  string newring="
  ring nring=("+charstr(basering)+"),("+va+","+pa+"),lp;";  
  string quotring="
  ring quring=("+charstr(basering)+","+pa+"),("+va+"),lp;";
  return(newring,quotring);
}

   ring s=(0,x),(a,b,c,d,e,f,g),dp;
   ideal i=x*b*c,d^2,f-g;
   i=std(i);
   def Q=basering;
   list l= prepareQuotientring(i);
   l;
   execute (l[1]);
   basering;
   execute (l[2]);
   basering;
   setring Q;