Let $a\_1,\ldots,a\_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such that
$$
s\_\lambda(a\_1,\ldots,a\_n) \ge 0
$$
for every partition $\lambda = (\lambda\_1 \ge \ldots \ge \lambda\_n)$, where $s\_\lambda$ denotes the [Schur polynomial](https://en.wikipedia.org/wiki/Schur_polynomial) associated to $\lambda$. **Do these inequalities imply that $a\_i \in \mathbb{R}\_+$ for all $i$?**


Here's what I know so far:


* The converse is obvious: if $a\_i \in \mathbb{R}\_+$ for all $i$, then the sum over Young tableaux formula immediately shows $s\_\lambda(a\_1,\ldots,a\_n) \ge 0$.
* It's enough to show that the $a\_i$ are real. Nonnegativity then follows by Descartes' rule of signs upon noting that the elementary symmetric functions $s\_{(1,\ldots,1)}$ are, up to alternating signs, the coefficients of the polynomial $\prod\_i (x-a\_i)$.
* The statement is true and easy to prove for $n = 2$: it's enough to postulate nonnegativity merely for $\lambda = (n,0,\ldots)$, i.e. on the complete homogeneous symmetric polynomials. On these,
$$
s\_{(n,0,\ldots)}(r e^{i\theta},r e^{-i\theta}) = \frac{(r e^{i\theta}))^{n+1} - (r e^{-i\theta})^{n+1}}{re^{i\theta}-re^{-i\theta}} = r^n \frac{\sin((n+1)\theta)}{\sin(\theta)},
$$
which for every nonzero $\theta$ is obviously negative for some $n$, proving the contrapositive statement.
This suggests that some Fourier analysis may be useful for the general case, or perhaps the [Harish-Chandra-Itzykson-Zuber formula](https://terrytao.wordpress.com/2013/02/08/the-harish-chandra-itzykson-zuber-integral-formula/), but I don't know how to go about it.



Nonnegativity locus of Schur polynomials
The answer to the main question is affirmative. The crucial result is due to M. Aissen, I. J. Schoenberg, and A. Whitney, [*J. Analyse Math.* **2** (1952), 93—103](https://link.springer.com/article/10.1007/BF02786970). For further details see the solution to Exercise 7.91(e) in *Enumerative Combinatorics*, vol. 2. (For some interesting information on A. Whitney, see [Biographical information on Anne Marie Whitney](https://mathoverflow.net/questions/320906/biographical-information-on-anne-marie-whitney))


