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8. Zeros of specific functions


    1. Add remark

      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 $n$.


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          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 \[P_n(z,w) = \sum_{k=0}^n \binom{n}{k}z^kw^{k(n-k)}\]

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              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 $|w|^2$).


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                  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.