TITLE: Is any submanifold of $\mathbb{R}^{n}$ the zero set of some polynomial?
QUESTION [5 upvotes]: For a given circle we have a corresponding equation to generate it. For a given ellipsoid we also can write a corresponding equation for it. In general, can we write for any given manifold an equation to characterize it? I googled this but I did not get useful information.

REPLY [2 votes]: No, certainly not.  Consider the curve $\{ (x,y): y = e^x \}$.  This is a submanifold of $\mathbb{R}^2$, but it is not the zero set of a polynomial equation.