TITLE: How can I prove that $(n-1)$-dimensional manifold is not contained in a $(n-2)$-dimensional affine variety?
QUESTION [5 upvotes]: I am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\mathbb{R}^n$. Let $W $ be an algebraic set defined by homogeneous forms in $\mathbb{R}[x_1, ..., x_n]$ for which $W \subseteq \mathbb{A}_{\mathbb{\mathbb{C}}}^n$ has dimension $n-2$ (as an algebraic set in $\mathbb{C}^n$). I want to prove that 
$$
M \not \subseteq W \cap \mathbb{R}^n. 
$$  
I think one reason why I am having trouble proving this (even though I suspect it might be quite basic) may be that I don't have a good understanding on how the dimension of the manifold and dimension of algebraic sets relate to one another. I would greatly appreciate any comments or references. Thank you.

REPLY [4 votes]: You seem to be asking whether it is possible that the topological dimension of the set of real points of an algebraic variety is greater than its algebraic dimension (in your case, the former is $n-1$ and the latter is $n-2.$) It seems to be a standard fact that this is impossible (inequality in the opposite dimension is quite possible, as in the standard example of $x^2+y^2 = -1.$) The standard reference seems to be:
Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics 10. Berlin: Springer (ISBN 3-540-33098-4/hbk). x, 662 p. (2006). ZBL1102.14041.