I've got a question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots \in \mathbb{Z}[X]$, where $f_0=1$. Now, define the following sequence of symmetric polynomials on variables $x_1, \ldots x_n$:
$$P_m(x_1, \ldots, x_n)=\sum\limits_{m_1+\cdots+m_n=m}f_{m_1}(x_1) \cdots f_{m_n}(x_n)$$
If you want, you can think of this as the coefficient of $y^m$ in the two-variable generating function $\prod\limits_{i=1}^{n}\left(\sum\limits_{j \ge 0}f_j(x_i)y^j\right)$. Now my question is, if you know $x_1, \ldots, x_n$ are positive integers, and you know the values of $P_m(x_1, \ldots, x_n)$ for $m=0, 1, \ldots$, what conditions need to be true on $f_1, f_2, \ldots$ in order for the values of of $x_1, \ldots, x_n$ to be completely determined (up to ordering)? I think it's key that we need to work with inputs in positive integers - if we work over $\mathbb{C}$, the answer may be different (see at the bottom).
For example, if $f_1(x)=x$, and $f_2=f_3=\cdots=0$, then $P_m$ is just the $m$-th elementary symmetric polynomial $\sigma_m$, and then obviously, $x_1, \ldots, x_n$ are determined by $\sigma_1, \sigma_2, \ldots$, being the roots of the polynomial with coefficients $(-1)^i\sigma_i$. If $f_i(x)=x^i$ for each $i$, then it's a little less straightforward, but still not hard: $P_m=\sum\limits_{j=1}^{m}(-1)^{j-1}\sigma_j P_{m-j}$, and so we can show by induction on $m$ that $P_1, \ldots, P_m$ together determine $\sigma_1, \ldots, \sigma_m$, and hence, $x_1, \ldots, x_n$ are again determined. For the general case, we can equivalently ask whether the values of $P_1, P_2, \ldots$ uniquely determine the values of $\sigma_1, \sigma_2, \ldots$ (and this is perhaps a more natural question).
As long as the polynomials $f_1$ aren't all constants, I don't, off the top of my head, know any sequences of polynomials $f_1, f_2, \ldots$ for which the values of $P_1, P_2, \ldots$ don't determine $x_1, \ldots, x_n$. So I'm wondering if it's true given that at least one of the $f_i$'s is nontrivial.
I feel it's worth pointing out that what I'm asking isn't the same thing as asking that the $P_i$ generate the ring of symmetric polynomials. For example, if the values of the $P_i$ were to determine the values of $\sigma_1^2, \sigma_2^2, \ldots$, that would determine the values of $\sigma_1, \sigma_2, \ldots$, even if the $P_i$ don't generate the ring. This is why the condition that the $x_i$'s are positive integers is important. That said, I don't know the answer over $\mathbb{C}$ either.
Anyone have any ideas? If it's of any relevance, in the student's research, the polynomial $f_i$ has degree $2i$ for each $i$.
