Trying to understand non-singulr points of an algebraic set in the reals

Question

Let $G(\mathbf{x}) \in \mathbb{Q}[x_1, \ldots, x_n]$ be a degree $d$ homogenous form. I am interested in the algebraic set $V(G) = \{ \mathbf{x} \in \mathbb{R}^n: G(\mathbf{x}) = 0 \}$.

Suppose $\mathbf{x}_0 \in V(G)$ is a non-singular point.

1) Does this mean that $V(G)$ in a small neighborhood of $\mathbf{x}_0$ is like a $(n-1)$-dimensional space? (I think this is true...)

2) How can I see this?

I would greatly appreciate any explanation. Thank you very much!

Answer 1

Yes, and this is essentially the implicit function theorem.