# How special are the polynomials amongst the smooth functions?

This is a naive question, so perhaps the answer will be made obvious by the right remark.

On a smooth manifold, there is no notion of polynomial (apart from constants). I would like to know if, nevertheless, some subalgebras of functions are more polynomial than others. I am only interested in local statements so I'll state the question for a neighbourhood of the origin in $\mathbb{R}^n$.

Consider the space of germs of smooth functions at zero in $\mathbb{R}^n$. The polynomial germs form a finitely-generated subalgebra of the algebra of smooth germs. Applying a diffeomorphism to a neighbourhood of zero, which we assume fixes zero, yields a different 'polynomial' subalgebra given by the new coordinates. Consider all the subalgebras that can be realised this way. Can these subalgebras be characterised within the algebra of smooth germs using just algebraic and topological properties, i.e. in a coordinate-free way?

Equivalently, I would like to know, if I have a finitely-generated unital subalgebra of the smooth germs that also possibly satisfies algebraic and topological conditions, does there necessarily exist a diffeomorphism that makes my subalgebra into polynomials (or maybe a subalgebra of the polynomials)? I am interested in statements generally along these lines, so please add anything you feel is relevant.

• A side note: the coordinate-free way of talking about $N$-variable polynomials is $S(V*)$ - the symmetric algebra of a dual of an $N$-dimensional vector space. Probably considering the case $V = T_pM$ (the tangent space to a point of a manifold) and somehow extending this algebra into some neighbourhood of the point would be helpful. – lisyarus Jun 3 '17 at 15:26