1 | initial version |

It is easy to compute what you want in Sage if you have a specific ring and specific number of variables. The method you want is the `vector_space_dimension`

method of the ideal I. Here is an example from the documentation for that method that involves the field `QQ`

(rational numbers) and two generators `u, v`

:

```
sage: R.<u,v> = PolynomialRing(QQ)
sage: g = u^4 + v^4 + u^3 + v^3
sage: I = ideal(g) + ideal(g.gradient())
sage: I
Ideal (u^4 + v^4 + u^3 + v^3, 4*u^3 + 3*u^2, 4*v^3 + 3*v^2) of Multivariate Polynomial Ring in u, v over Rational Field
sage: I.dimension()
0
sage: I.vector_space_dimension()
4
```

2 | No.2 Revision |

It is easy to compute what you want in Sage if you have a specific ~~ring ~~field `k`

and specific number of ~~variables. ~~generators. The method you want is the `vector_space_dimension`

method of the ideal I. Here is an example from the documentation for that method that involves the field `QQ`

(rational numbers) and two generators `u, v`

:

```
sage: R.<u,v> = PolynomialRing(QQ)
sage: g = u^4 + v^4 + u^3 + v^3
sage: I = ideal(g) + ideal(g.gradient())
sage: I
Ideal (u^4 + v^4 + u^3 + v^3, 4*u^3 + 3*u^2, 4*v^3 + 3*v^2) of Multivariate Polynomial Ring in u, v over Rational Field
sage: I.dimension()
0
sage: I.vector_space_dimension()
4
```

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