TITLE: Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space
QUESTION [9 upvotes]: Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ for $v\in V$. 
A different way to look at it is the following: Consider the polynomial ring $R=\mathbb{C}[x_1,\ldots,x_n]$ and $f$ a homogeneous polynomial of degree $d$: Then, I want to show that there are linear polynomials $h_1,\ldots,h_k$ such that $f$ is a linear combination of the $d$-th powers $h_i^d$.
In the case $d=2$, this follows from $2xy = (x+y)^2 - x^2 - y^2$. For higher $d$, I recall seeing a proof involving multinomial coefficients once, but I do not remember the details. I have tried to work it out again, but it seems a bit cumbersome, so I am asking whether you know any textbook where this result is proved. If you know an easy proof, I'd be very happy if you could outline it, though.

REPLY [2 votes]: In my answer here I note that symmetric tensors, as multilinear functionals, descend to linear maps on the symmetric power of the underlying vector space. I then reason that if we could show that $\mathrm{Sym}^n V$ is generated by $n$th powers of elements from $V$ the question on tensors would then be answered in its general form decisively. I remark that this is formally equivalent to the elementary symmetric polynomials $e_n$ being expressible as sums of $n$th powers of homogeneous polynomials.
This was the subject of my question here, which received a correct answer (containing a proof of the claim) from user m_l. It was very combinatorial and indeed involved multinomial coefficients, though I'm not sure how related it is to what you've seen before. (Unfortunately, at this point in time I am the only person to have upvoted poor m_l.) It requires the characteristic of the base field be greater than the power $n$ in question (or zero, of course).