The following problem arose in collaborative work with Subhro Ghosh:

**Question**: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
$L_n=n^{-1}\sum \delta_{z_i}$ (a probability measure on the complex plane).
Let ${\cal M}$ denote the collection of limit points (in the weak topology)
of those empirical measures obtained when all $a_i$
are constrained to be non-negative reals (but otherwise, arbitrary $a_i$), with limit points being probability
measures. Can ${\cal M}$ be characterized?

**Remarks:** of course, probability measures in $\cal M$ are symmetric about the real axis, so enough to look at restriction of measures to the (closed) upper half plane. Probability measures supported on the negative half plane belong to $\cal M$. On the negative side,
by a theorem of Obrechkoff, any probability measure in $\cal M$ charges the symmetric cone of total width $\alpha$ around the real axis with mass at most $\alpha/\pi$. Finally, for random polynomials with $a_i$ iid with bounded moments of some order, the limit point is the uniform measure on the circle, and from this (or using products of polynomials $\sum_{i=0}^k z^i$) one easily shows that any radially symmetric probability measure belongs to $\cal M$. Results of Barnard et als on factoring polynomials with positive coefficients allow one to construct further examples of measures in $\cal M$.
There are more examples, but we look for a characterization of $\cal M$.