4.5 Parameters

Let us now deform a given 0-dimensional ideal j 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.

For almost all a in Q this is the same as dim_Q Q[x,y]/j0, where j_0=j_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