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 $\begin{align*} &x^4,y^4,z^4,w^4,(x+y+z+w)^4, (x+y+z)^4, (x+y+w)^4, (x+w+z)^4, \\ &(w+y+z)^4, (x+y)^4, (z+y)^4, (w+y)^4, (x+w)^4, (w+z)^4,\text{ and }(x+z)^4\,? \end{align*}$