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Sorry if this is a stupid question to ask. This is an exercise from Gathmann's Algebraic Geometry.

Show that every affine variety in the real affine space $A^n_R$ is the zero locus of one polynomial.

If some context might help, he is talking about dimension of a topological spaces in that section. In the same problem, he asks to show that every Noetherian topological space is compact, which is easier to show.

I have no idea where to start. Please give me some hint. Thank you so much!

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    $\begingroup$ $\sum_i f_i^2 =0 \iff f_i=0 \ \forall i$. $\endgroup$ – user64687 Feb 20 '15 at 11:45
  • $\begingroup$ That is a great hint. Thank you. $\endgroup$ – KittyL Feb 20 '15 at 11:51
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By definition (I assume this is your book's definition-- I don't really like using the language of classical algebraic geometry when I am working over $\mathbb{R}$), the variety is the vanishing locus of $m$ polynomials $f_1, \ldots, f_m$. So we need to construct ONE polynomial that vanishes if and only if all $m$ of those polynomials vanish. This we can do using a trick:

$$ g(x_1, \ldots, x_n) = f_1(x)^2 + \ldots f_m(x)^2 $$

This clearly vanishes if all the $f_i$ vanish; if they don't all vanish, it's clearly positive and in particular not zero.

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  • $\begingroup$ That is a very simple proof. Thank you! $\endgroup$ – KittyL Feb 20 '15 at 11:50

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