What is the connection between the complex variety and real variety? The complex variety and real variety are defined by the same set of polynomial $(f_1, f_2,...f_n)$ except the solution set one is defined over $\mathbb{C}$, the other is defined over $\mathbb{R}$. I know the dimension and degree of complex variety. The question is what are the dimension and degree of real variety. Any theorem, book, note relates this would be great.
This question arises when I solve a set of polynomial equations, I can use numerical algebraic geometry to get complex solution(complex variety) of this set of polynomial equations, but I really want is real solution of this set of polynomial equations(real variety). Right now, I have the dimension and degree of complex solution(complex variety), I just want to know the dimension and degree of the real variety?
 A: The only general relation between the dimension of the real points and the complex points of a variety is that the former is not more than the latter. Depending on your definition of "degree" (for instance, number of intersections counted appropriately with a generic complementary dimensional linear subspace), the notion may be worthless. This is one reason to upgrade to schemes - your theory behaves correctly over fields that aren't algebraically closed, and you can still answer questions like "does this have a real point" by descent theory and the like.

If you're looking for explicit real solutions of a system of polynomial equations, the best tools I'm aware of come from the area of semi-algebraic geometry. Specifically, what you'll want to do is compute a cylindrical algebraic decomposition of your set. This gives you a decomposition of the solution set as a finite union of sets which are semialgebraically isomorphic to $(0,1)^d$ and if two of these cells intersect after a coordinate projection, their images are equal. For an introduction, I like these notes by Michel Coste; if you're looking to compute a specific example, Mathematica has an implemenation as does SAGE, though I have never used either. (I've done work in semi-algebraic geometry, but my work almost never relies on computing a specific decomposition, so I'm not super familiar with the current bleeding edge of the implementation of the algorithm - just the existence is enough for my purposes most of the time).
