Zeros of polynomials with real positive coefficients

Question

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 $\cal M$.

Answer 1

I don't have a complete proof yet, but I have a plausible conjecture. Let $\mu$ be a probability measure in the plane, define the potential $$u(z)=\int\log|1-z/t|d\mu(t).$$ Then I conjecture that $\mu\in M$ iff $u$ satisfies $u(z)\leq u(|z|)$ for all $z$.

It is evident that this condition is necessary. It seems that it is strictly stronger than the Obrechkoff condition. I don't think that this condition can be restated as a simple property of $\mu$ itself.

To prove the sufficiency, I am first going to restrict to a dense subclass of $\mu$ with convenient properties (it is clear that it is enough to prove sufficiency for a dense subclass). The convenient properties I have in mind is that $\mu$ does not charge some small angular sector $|\arg z|<\epsilon$ and that it behaves nicely near $0$ and $\infty$, say has some small atom at $-\epsilon$ and nothing else in the disc $|z|<100\epsilon$, and similarly at infinity. In addition, I want to require that $u(|z|)>u(z)$ for all $z$ except on the positive ray.

Then I am going to discretize the measure to obtain a polynomial, whose $(1/n)\log|P_n|$ approximates $u$ nicely near the positive ray, and apply the saddle point method to the integral $$\int_{|z|=r}\frac{P_n(z)}{z^k}\frac{dz}{z},$$ with $n\to\infty$, using the nice behavior near the positive ray, and obtain an asymptotic for the coefficients which will show that they are positive.

The difficulty is that the asymptotics must be uniform in $k$, but I hope to achieve this by the arrangement near $0$ and $\infty$ described above.

In fact, there is an (unpublished and unproved) conjecture of Alan Sokal that if a polynomial satisfies $|P(z)|<P(|z|)$ then some sufficiently high power has positive coefficients. This of course would imply sufficiency of my condition.

ADDED on July 19. The above outline is correct; we are writing a proof which will soon be posted on arxiv.

ADDED on August 23. Here is the precise statement. A probability measure $\mu$ is a limit measure if and only if it is symmetric with respect to complex conjugation, and $u(z)\leq u(|z|)$ where $$u(z)=\int_{|\zeta|\leq 1}\log|z-\zeta|d\mu(\zeta)+\int_{|\zeta|>1}\log|1-z/\zeta|d\mu(\zeta).$$ (The potential I wrote earlier may be divergent for some probability measures, so it has to be modified a little bit). A proof of this is now available: https://arxiv.org/abs/1409.4640.

UPDATE on September 10, 2014. What I called "Sokal's Conjecture" above (Theorem 1 in the preprint cited above) turned out to be known before. It was proved by V. de Angelis, MR1976089. This was found as a result of David Handelman's answer to another MO question: Stability of real polynomials with positive coefficients.