# Can any smooth manifold be realized as the zero set of some polynomials?

Is any real smooth manifold diffeomorphic to a real affine algebraic variety? (I.e. is there an "algebraic" Whitney embedding theorem?)

And are all possible ways of realizing a manifold $M$ as an algebraic variety equivalent? I.e. suppose $M$ is diffeomorphic to varieties $V_1$ and $V_2$, are these isomorphic in the algebraic category?

Admittely I'm just asking out of curiosity after reading this question: Can manifolds be uniformly approximated by varieties?

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The answer to the second question is certainly not: just take two elliptic curves with slightly different $j$-invariants.