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A.6 Parameters
Let us deform the above ideal now by introducing a parameter t
and compute over the ground field Q(t).
We compute the dimension at the generic point,
i.e.,
![$dim_{Q(t)}Q(t)[x,y]/j$](sing_165.png){kind=small}
.
(This gives the
same result as for the deformed ideal above. Hence, the above small
deformation was "generic".)
For almost all
this is the same as
![$dim_Q Q[x,y]/j_0$](sing_167.png){kind=small}
,
where
.
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 |