Let $n$ be a positive integer and $W\_n$ be the linear subspace of the real vector space $\mathbb{R}[X\_1,\cdots,X\_n]$ generated by the following set
$$S\_n=\{X\_1^{i\_1}\cdots X\_n^{i\_n}:i\_1+\cdots+i\_n=n\ \text{and}\ i\_k\in \{0,1,2\},k=1,\cdots,n\}.$$
Let 
$$T\_n=\{(X\_{i\_1}-X\_{i\_2})(X\_{i\_2}-X\_{i\_3})\cdots(X\_{i\_n}-X\_{i\_1}):i\_1,\cdots,i\_n\ \text{is a permutation of}\ 1,\cdots,n\}.$$
It is easy to see that $T\_n\subset W\_n$.


When $n$ is odd, the expansion of any polynomial in $T\_n$ does not contain the monomial $X\_1X\_2\cdots X\_n$. So $T\_n$ can not generate $W\_n$.


When $n=2m\geq 8$ is even, we have 
$$|S\_n|=\sum\limits\_{k=0}^m\binom{n}{2k}\binom{2k}{k}<
\dfrac{1}{2}(n-1)!=|T\_n|.$$
So I want to ask can $W\_n$ generated by $T\_n$ when $n\geq 8$ is even?



A linear subspace of $\mathbb{R}[X_1,\cdots,X_n]$ and its generated set
$\let\eps\varepsilon$Substitute $X\_k\leftarrow\eps\_k:=\exp(2\pi ki/n)$. Notice that $\arg(\eps\_k-\eps\_\ell)\equiv (\pi+\arg \eps\_k+\arg\eps\_\ell)/2\pmod{\pi}$. Summing up, we get that the argument of every element of $T\_n$ is congruent to a constant modulo $\pi$; so all their real lnear combinations also share this property. On the other hand, the elements of $W\_n$ may have $n$ distinct arguments. This works for all $n>2$, regardless of the parity.


**NB.** Surely, the fact is still true over $\mathbb C$, since the elements of $S\_n$ are necessarily rational linear combinations of those in $T\_n$, provided that they are complex such combinations.


