Let $f(x_1,...,x_n)$ be a symmetric polynomial, so that $f(x_{\sigma(1)},...,x_{\sigma(n)}) = f(x_1,...,x_n)$ for each $\sigma \in S_n$. For $1 \leq k \leq n$, we denote by $s_k$ the $k^{\text{th}}$ elementary symmetric function given by $s_k(x_1,...,x_n) = \sum_{1 \leq j_1 < ... < j_k \leq n}x_{j_1}\cdots x_{j_k}$. One is asked to prove by induction on $n$ that $f(x_1,...,x_{n-1},0) = g(s_1(x_1,...,x_{n-1},0),...,s_{n-1}(x_1,...,x_{n-1},0))$ for some polynomial g in $n-1$ variables.
The result is trivial for $n = 1$ since $f(0)$ is a constant polynomial. Now, assuming that the result holds for $n = k$, where $k \geq 1$ is fixed, we can write $f(x_1,...,x_k, x_{k+1}) = \sum_{j = 0}^mg_j(x_1,...,x_k)x_{k+1}^j$, where $g_j(x_1,...,x_k)$ is a symmetric polynomial for $0 \leq j \leq m$. Hence $f(x_1,...,x_k,0) = g_0(x_1,...,x_k)$. It is not immediately obvious to me how the induction hypothesis leads to an expression of the form $f(x_1,...,x_k,0) = g_0(x_1,...,x_k) = g(s_1(x_1,...,x_k,0),...,s_k(x_1,...,x_k,0))$, where $g$ is some polynomial in $k$ variables. Any hints/suggestions would be much appreciated.
