The ideal below is zero-dimensional, so the set of solutions are points in C[x,y,z]. The intersection of F with each variable is a univariate polynomial so there are no algebraically independent variables.
>
|
|
| (1) |
>
|
|
| (3) |
>
|
|
The first two equations generate a curve in C[x,y,z]. All of the variables are algebraically independent.
>
|
|
>
|
|
Over GF(2) the situation is different, z is algebraically independent so the ideal generates a "curve".
>
|
|
>
|
|
| (9) |
>
|
|