$\newcommand{\Cat}{{\rm Cat}}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\freestar}{ \framebox[7pt]{\star} }$

## 8. Zeros of specific functions

1. #### Conjecture 8.1.

[A. Vishnyakova] Let $p_n(z) = \sum_{k=0}^n (-1)^k(2k+1)z^{k(k+1)}$ where, $n=2,3,4,\ldots.$ Each $p_n$ has no real zeros for even $n$, and exactly two real zeros for odd $n$.
• #### Problem 8.2.

[A. Vishnyakova] Consider the function $f(z) := \sum_{k} \frac{z^k}{(a^k+1)(a^{k-1}+1)\cdots (a+1)}, \qquad a>1.$ For which values of $a$ is $f$ in $\mathcal{L}\text{-}\mathcal{P}$?
•     Let $P_n(z,w) = \sum_{k=0}^n \binom{n}{k}z^kw^{k(n-k)}$

#### Conjecture 8.3.

[A. Sokal] For all $|w|>1$, all the zeros of $P_n(\cdot,w)$ are simple and separated in modulus (by a factor of at least $|w|^2$).
• #### Problem 8.4.

Generalize conjecture conj_sokal to $Q_n(z,w)=\sum_{k=0}^n a_kz^kw^{k(n-k)}$. For what class of $\{a_n\}$ does the conjecture still hold?

Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.