//SINGULAR Procedure4.6.1





proc primaryTest(ideal i, poly p)

"USAGE:   primaryTest(i,p); i standard basis with respect to
          lp, p irreducible polynomial in K[var(n)], 
          p^a=i[1] for some a;
ASSUME:   i is a zero-dimensional ideal.
RETURN:   an ideal, the radical of i if i is primary and in 
          general position with respect to lp, 
          the zero ideal else.
EXAMPLE:  primaryTest; shows an example
"  
{
   int m,e;
   int n=nvars(basering);
   poly t;
   ideal prm=p;

   for(m=2;m<=size(i);m++)
   {
     if(size(ideal(leadexp(i[m])))==1)
     {
       n--;
//----------------i[m] has a power of var(n) as leading term
       attrib(prm,"isSB",1);
//--- ?? i[m]=(c*var(n)+h)^e modulo prm for h 
//     in K[var(n+1),...], c in K ??
       e=deg(lead(i[m]));
       t=leadcoef(i[m])*e*var(n)+(i[m]-lead(i[m]))
       /var(n)^(e-1);
       i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m];
//---if not (0) is returned, else c*var(n)+h is added to prm
       if (reduce(i[m]-t^e,prm,1) !=0)
       {
         return(ideal(0));
       }
       prm = prm,cleardenom(simplify(t,1));
     }
   }
   return(prm);
}

   ring s=(0,x),(d,e,f,g),lp;
   ideal i=g^5,(x*f-g)^3,5*e-g^2,x*d^3;
   primaryTest(i,g);