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A.1.5 Parameters

Let us deform the ideal in Long coefficients by introducing a parameter t and compute over the ground field Q(t). We compute the dimension at the generic point, i.e., $dim_{Q(t)}Q(t)[x,y]/j$.(This gives the same result as for the deformed ideal above. Hence, the above small deformation was "generic".)

For almost all $a \in Q$this is the same as $dim_Q Q[x,y]/j_0$,where $j_0=j\vert _{t=a}$.

 
  ring Rt = (0,t),(x,y),lp;
  Rt;
==> //   characteristic : 0
==> //   1 parameter    : t 
==> //   minpoly        : 0
==> //   number of vars : 2
==> //        block   1 : ordering lp
==> //                  : names    x y
==> //        block   2 : ordering C
  poly f = x5+y11+xy9+x3y9;
  ideal i = jacob(f);
  ideal j = i,i[1]*i[2]+t*x5y8;  // deformed ideal, parameter t
  vdim(std(j));
==> 40
  ring R=0,(x,y),lp;
  ideal i=imap(Rt,i);
  int a=random(1,30000);
  ideal j=i,i[1]*i[2]+a*x5y8;  // deformed ideal, fixed integer a
  vdim(std(j));
==> 40