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Is there a system $\{s_1, \cdots, s_m\}$ of symmetric polynomials of $z_1, \cdots, z_n \in \mathbb{C}$ such that

$$s_1(z_1, \cdots, z_n) = c_1$$ $$s_2(z_1, \cdots, z_n) = c_2$$ $$\cdots$$ $$s_m(z_1, \cdots, z_n) = c_m$$

has at most one solution $(z_1, \cdots, z_n)$ up to permutation, for all choices $c_1, \cdots, c_m \in \mathbb{C}$?

If so, what is the minimum value of $m$? And what are all such systems of polynomials $\{s_1, \cdots, s_m\}$ with this property?

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This is true whenever the $s_i$ generate the ring of symmetric functions since if you know the elementary symmetric polynomials $e_1, ... e_n$ in the $z_i$ you can write down a polynomial they satisfy. This is true, in particular, if the $s_i$ are

I know that the second can be replaced with "any $n$ consecutive power symmetric polynomials," and I would expect the third can as well. I don't know about more general characterizations; that seems like a hard problem. I think the minimum $m$ should be $n$.

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  • $\begingroup$ You mean $m$ is at most $n$, right? $\endgroup$ Mar 12, 2011 at 17:37
  • $\begingroup$ No. You want at most one solution, right? So there should be more constraints, not less. $\endgroup$ Mar 12, 2011 at 17:42
  • $\begingroup$ I'm not sure we have the same question in mind. I ask for the minimum $m$ such that there exists as system $s_1,\cdots,s_m$ that determines a unique solution. On other words, if I know $z_1, \cdots, z_n$, and I tell my friend the values of $s_1, \cdots, s_m$ evaluated at $(z_1, \cdots, z_n)$, then my friend can work out $(z_1, \cdots, z_n)$. $\endgroup$ Mar 12, 2011 at 17:58
  • $\begingroup$ The minimum should be exactly $n$. That's exactly what is meant by the statement that $m$ is at least $n$. $\endgroup$ Mar 12, 2011 at 18:02

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