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Let $V$ be any irreducible variety over $\mathbb{R}$, prove that if $dim_{\mathbb{R}}V(\mathbb{R})= dim(V)$, then $V(\mathbb{R})$ is dense in $V$.

Any hints to make a start?

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    Dear CC, How do you conclude that $\dim_{\mathbb R}V(\mathbb R) = 1$. What if $V$ were the affine plane? Regards,2012-10-19
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    Why is $\dim_{\Bbb R}V(\Bbb R)=1$? Isn't $\Bbb A^2(\Bbb R)$ two-dimensional?2012-10-19
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    Oh yes.. I think I misunderstood the definition of dimension of an algebraic variety.2012-10-19
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    What do you mean by $\dim_{\mathbb{R}} V(\mathbb{R})$? $V(\mathbb{R})$ isn't necessarily, say, a manifold, so it's not clear (to me) what notion of dimension is meant here.2012-10-19
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    @Matt: it's not clear to me that each component will have the same dimension, though. Do we take a supremum over all components?2012-10-19
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    Dear CC, How about some context? What do you know? Regards,2012-10-19
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    @QiaochuYuan: Dear Qiaochu, That's I would do (at least, the exercise is true with that interpretation!). Regards,2012-10-19
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    @QiaochuYuan In fact, we don't have such definition in our lecture note (We always have HW assignments like this that contain unknown notations..). However, I guess since $V$ is determined by a prime ideal $P$, does $dim_R(V(R))$ means the kul dimension of $P$ in the ring $\mathbb{R}[x_1,..,x_n]$?2012-10-19
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    @MattE I'm currently seraching notations from the book "Algebraic Curves, Algebraic Manifolds and Schemes" by Danilov and Shokurov.2012-10-19
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    @CC_Azusa: I assumed that that was what $\dim(V)$ meant.2012-10-19
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    Dear CC, No, $\dim_{\mathbb R}V(\mathbb R)$ means the topological dimension of the real points. Regards,2012-10-19
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    @MattE Thanks Matt, that makes more sense2012-10-19

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This is not a solution, but is more of an extended comment:

You might want to consider some examples before trying to prove this in general. Here are some that you can try (in some of the them the hypothesis of your problem holds, in others it doesn't):

(1) $V$ is the affine curve given by the equation $x^2 + y^2 = 0$.

(2) $V$ is the affine curve given by the equation $y^2 = x^2(x-1)$.

(3) $V$ is the affine curve given by the equation $y^2 = x^3 - x$.

In each case you should plot the real points and see what dimension they are, and then consider whether or not they are Zariski dense in $V$ (i.e. in the set of complex points).