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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!

KittyL
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1 Answers1

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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.

hunter
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  • That is a very simple proof. Thank you! – KittyL Feb 20 '15 at 11:50
  • Feynman in his celebrated three-volume course on physics makes fun of people who want to reduce physics to as few equations as possible by remarking, using this old trick, that one single equation already suffices. (The $f_i$'s needn't be algebraic, of course, in his context). – Georges Elencwajg Apr 18 '22 at 09:06