How does one define a dimension of $V(F) = \{F = 0\}$ over the reals? I have questions about how dimension is defined for an affine varieties restricted to $\mathbb{R}^n$. Let $F(\mathbf{x}) \in \mathbb{R}[x_1, ..., x_n]$. 
How does one define dimension for $V_{\mathbb{R}}(F) = V(F) \cap \mathbb{R}^n= \{ \mathbf{x} \in \mathbb{R}^n: F(\mathbf{x}) = 0  \}$? 
One way to define this is to use the subspace topology (as a subspace of $\mathbb{A}_{\mathbb{C}}^n$) and define it 
as the length of a maximal chain of irreducible closed sets (here closed sets are algebraic sets in $\mathbb{C}^n$ intersected with $\mathbb{R}^n$). 
Another way I think is as the length of a maximal chain of irreducible algebraic sets (zero sets of real polynomials) in $\mathbb{R}^n$. 
They both seem like a natural definition to me... Are these two possible definitions always give the same quantity if we are considering an algebraic set (in $\mathbb{R}^n$) defined by real polynomials?
Thank you very much. 
 A: It is always true that the scheme-theoretic dimension of $\operatorname{Spec}\Bbb R[x_1,\cdots,x_m]/(f_1,\cdots,f_k)$ is the Krull dimension of $\operatorname{Spec}\Bbb R[x_1,\cdots,x_m]/(f_1,\cdots,f_k)$ as a ring, which is equal to the Krull dimension of $\Bbb C[x_1,\cdots,x_m]/(f_1,\cdots,f_k)$ as a ring which is equal to the scheme-theoretic dimension of $\operatorname{Spec}\Bbb C[x_1,\cdots,x_m]/(f_1,\cdots,f_k)$. This may be seen by considering what $-\otimes_{\Bbb R} \Bbb C$ does to chains of ideals.
On the other hand, the dimension of the $\Bbb R$ points need not behave so nicely. For $X\subset \Bbb A^n_{\Bbb R}$, the set of real points may have any dimension between $0$ and $n$: consider the variety cut out by $\sum_{i=1}^d x_i^2$.
One saving grace is that if $X$ is a smooth $\Bbb R$ variety, then the dimension of the real points of $X$ as a manifold is either the algebraic dimension of $X$ or the set of real points of $X$ is empty. This may be seen by considering the Jacobian criteria for smoothness and the implicit function theorem. One can then upgrade this to say that a general variety has the expected dimension of it's real points if you can find a smooth point somewhere by considering $X^{sm}\subset X$.
Unfortunately, I've found that there's not even much standardized terminology about the mismatch of these dimensions - I searched diligently and asked a question on this website nearly 3 years ago about the matter and it got no responses.
