Consider the polynomials $f,g,h \in \mathbb C [x_1, \dots,x_n]$ defined by
$ f(x)=x_1^2 + \cdots + x_{n-2}^2, \quad g(x)=x_{n-1},$ and $ h(x)=x_n.$
How can I compute the dimension of the variety
$$ V \subset \mathbb A_{\mathbb C}^n: f=g=h=0 ?$$
Any help would be really appreciated.
It seems you want to find out the Krull dimension of the ring $$\mathbb C [x_1, \dots,x_n]/(x_1^2 + \cdots + x_{n-2}^2,x_{n-1},x_n)\simeq\mathbb C [x_1, \dots,x_{n-2}]/(x_1^2 + \cdots + x_{n-2}^2)$$ which is $n-3$.