Do these special power functions generate all homogeneous symmetric polynomials? Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree. 
My question is as follows. To be succinct, let's say we have four variables. Are all homogeneous symmetric functions of a given a degree (let's say four) in $x,y,z,w$ generated over rational numbers by the special power functions given by $$\begin{align*}
&x^4,y^4,z^4,w^4,(x+y+z+w)^4, (x+y+z)^4, (x+y+w)^4, (x+w+z)^4, \\
&(w+y+z)^4, (x+y)^4, (z+y)^4, (w+y)^4, (x+w)^4, (w+z)^4,\text{ and }(x+z)^4\,?
\end{align*}$$
 A: I assume that by "generated over rational numbers" you mean "equal to a linear combination with rational coefficients". In this case the answer is no: for example, the symmetric function$$
p(x,y,z,w) = x^2y^2 + x^2z^2 + x^2w^2 + y^2z^2 + y^2w^2 + z^2w^2
$$
is not a linear combination of the 15 polynomials you listed. To see this, note that if $p$ did equal a linear combination of those 15 power functions, then that identity would remain valid when the specific values $y=0$, $z=0$, $w=1$ are substituted in. However, $p(x,0,0,1) = x^2$, while each of those 15 power functions equals one of $0$, $1$, $x^4$, or $(x+1)^4$ when these values are substituted in; and it's easy to verify that $x^2$ cannot be written as a linear combination of $0$, $1$, $x^4$, and $(x+1)^4$.
(Some symmetric functions, such as $xyzw$, can be written as a linear combination of those 15 power functions; so I don't know what general principle underlies this.)
A: If you have $n$ variables and look at the dimension $H(d)$ of the vector space of homogeneous symmetric polynomials of degree $d$, $H$ grows like a polynomial in $d$ of degree $n-1$.
Meanwhile, the number of "special power functions" you have is always $2^n-1$, and doesn't grow when $d$ grows. Thus for $n>1$ and a high enough degree, you can't generate all the homogeneous symmetric polynomials of degree $d$ only with the special power functions.
A: The answer is obviously no: since all polynomials you wrote down are homogeneous polynomials of degree $4$ with respect to the total degree, so they cannot be used to build anything with terms of total degree not divisible by $4$, like $xyzt+x+y+z+w$.
