//SINGULAR Example 2.8.10




ring R = (0,a,b,c,d),(x,y,z,u),(c,dp);
ideal E =  3x +   y +  z -  u - a,
          13x +  8y + 6z - 7u - b,
          14x + 10y + 6z - 7u - c,
           7x +  4y + 3z - 3u - d;

ideal EE =  E, x + y + z - u;
option(redSB);
simplify(std(E),1);             //compute reduced SB

std(EE);

ring R1 = 0,(x,y,z,u,a,b,c,d),(c,dp);
ideal EE = imap(R,EE);
std(EE);

LIB "matrix.lib";
proc coeffMat (ideal i)
{
   int ii;
   int n = nvars(basering);
   int m = ncols(i);
   matrix C[m][n];
   for ( ii=1; ii<=n; ii++)
   {
      C[1..m,ii] = i/var(ii);
   }
   return(C);
}

proc coeffMatExt(ideal i)
{
   matrix C = coeffMat(i);   
   C = concat(C,transpose(jet(i,0)));
   return(C);
}

setring R;
matrix CE = coeffMatExt(EE);
matrix C = coeffMat(E);

setring R1;
matrix CE = imap(R,CE);
std(transpose(CE));

ring R2 = 0,(x,y,z,u,a,b,c,d),(c,dp);
ideal p = 7a-2b+2c-4d;
qring qR= std(p);
matrix C = imap(R,C);
matrix CE= imap(R,CE);
matrix b[5][1]=CE[1..5,5];      //the r.h.s. of (**)
lift(C,b);