# Zero locus of a set of smooth functions. Smooth analogue of affine varieties.

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.

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.)
• 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. – Fallen Apart Mar 2 '18 at 13:49