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The notion of an affine variety is one of the most fundamental concepts in algebraic geometry. It is just the zero locus of some finite set of polynomials $f_1,\dots,f_k$ from $\mathbb{F}[X_1,\dots,X_n]$.

In the class of smooth functions on $\mathbb{R}^n$ we can do the same. If $ f_1,\dots,f_k$ are from $C^\infty(\mathbb{R}^n)$, then we can define $V(f_1,\dots, f_k)$ as the zero locus of $f_1,\dots,f_k$, i.e., $$V(f_1,\dots, f_k)=\lbrace x\in\mathbb{R}^n:f_i(x)=0, i=1\dots k\rbrace.$$

I am aware that $V(f_1,\dots, f_k)$ might not be a manifold, but still it is a very natural object to study.

Question. Do objects like $V(f_1,\dots, f_k)$ have some name?


Edit. The point of the question is not to classify $V(f_1,\dots, f_k)$, but just to have a concrete name for such objects.

$V(f)$ are usually know as level sets (or level surfaces/hypersurfaces). Similarly $V(f_1,\dots,f_k)$ can be seen as $V(F)$ where $F:\mathbb{R}^n\to\mathbb{R}^k$ is a smooth vector valued function. However, it looks like the notion of level set applies only to real valued functions.

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They are called closed subsets :-) Indeed, any closed set $E \subset \Bbb R^n$ can be realized as such objects, see here.

(In fact, you only need one function because $V(f_1, \dots, f_n) = V(f_1^2 + \dots + f_n^2)$ over the reals.)

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  • $\begingroup$ Good to know that fact, but the intension of my question was different (even thou I haven't specified that). Usually $V(f)$ is called a level set (or level surface/hypersurface). Then we can compute gradient of $f$ and see if some point is regular or critical. I wonder if $V(f_1,\dots, f_k)$ have some name. PS. I will update a question a little bit in a moment. $\endgroup$ – Fallen Apart Mar 2 '18 at 13:49
  • $\begingroup$ @FallenApart I see, then I think my answer was not so appropriate. I'll delete it soon :) $\endgroup$ – Nicolas Hemelsoet Mar 2 '18 at 13:52
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    $\begingroup$ Keep it! I found out something interesting from it and mayby others will do as well $\endgroup$ – Fallen Apart Mar 2 '18 at 13:57

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