Let $g_1 = x_1+x_2+x_3+x_4$, $g_2 = x_1^2+x_2^2+x_3^2+x_4^2$, $g_3= x_1x_2+x_3x_4$, $g_4 = x_1x_3+x_2x_4$, $g_5 = x_1^3+x_2^3+x_3^3+x_4^3$. I want to construct a polynomial $s(y_1,\cdots,y_5)$ such that $s(g_1,\cdots,g_5) = x_1 \cdots x_4$. Does somebody have an idea on how to do this?
What I have done so far: $s_1 = y_1$, $s_2=1/2(y_1^2-y_2)$, $s_3 = 1/6 y_1^3-1/2y_1 y_2+1/3y_5$ This gives the three elementary symmetric polynomials when pluging in $g_i$ for $y_i$ in $s_j$. Now how to proceed?