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8. Zeros of specific functions


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      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 8.3 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.