When a parametric represention is given ,

we can find the smallest affine variety contains its all points.

And in the books , There is a theorem(Elimination theorem) is used to solve this problem by considering the Groebner basis.

My problem is can I always eliminate those parameters ?

Here is a counter example

and I compute the Groebner basis by Maple in lex order , grlex order

,elimination order

and how can i deal with such examples if I want to find the smallest variety contains the parametrization?

thanks!!!