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Is there a name or notation for a Zariski closed set in $\mathbb{C}^n$ which when intersected with $\mathbb{R}^n$ has non-trivial dimension?

For example for $n=2$, if $f(x,y) = (x^2 + 1)(x^2 + y^2 - 1)$ then is there a name for sets like $\{(x,y) \in \mathbb{C}^2:f(x,y) = 0\} \cap \mathbb{R}^2 = \{(x,y) : x^2 + y^2 = 1\}$?

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Can you give us an example? –  Matt Jul 17 '11 at 21:02
    
Thanks! Some $n$'s appeared. –  Matt Jul 17 '11 at 21:52
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1 Answer

There is in fact a definition of such sets, they are called the set of real points of an algebraic set. Let $I \subset \mathbb{R}[x_1,\dots,x_n]$ a polynomial ideal, let $V(I)$ the algebraic set defined by $I$ over $\mathbb{C}$ and write $V_\mathbb{R}(I) = V(I)\cap \mathbb{R}^n$. The affine algebra $A = \mathbb{R}/I$ is called the $\mathbb{R}$-coordinate ring of $V_\mathbb{R}(I)$. We have the following theorem, known as the Weak Real Nullstellensatz

Theorem. $A$ is semireal if and only if $V_\mathbb{R}(I)\neq \emptyset$

The theorem is valid over a more general context, the notion of a real closed set wich is the starting point of Real Algebraic Geometry, you can check this theory on the book of Bochnak, Coste and Roy Real Algebraic Geometry.

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