Let $P\_n$ be the space of polynomials in $n$ variables over a field of characteristic 0. 
I'm very sure that the space spanned by powers of linear functions is the whole space $P\_n$.
Anyone can come up with a proof, or indicate a reference for it?
Thanks a lot.



Powers of linear functions span the space of polynomial functions?
Yes, actually, the space of homogeneous degree $d$ polynomials in $(n+1)$ variable is generated by 
$$(x\_0+i\_1x\_1+\ldots+i\_nx\_n)^d,\quad 0\le i\_1,\ldots,i\_n\le d.$$
For proof, write it as a binomial
$$[(x\_0+i\_1x\_1+\ldots+i\_{n-1}x\_{n-1})+i\_nx\_n]^d$$
and, using Vandermonde, get all products $(x\_0+i\_1x\_1+\ldots+i\_{n-1}x\_{n-1})^px\_n^q$, $p+q=d$. Then, use induction.


