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9. Miscellaneous Problems


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      Problem 9.1.

      [P. Gauthier] Let be K\subset\mathbb{C} be compact, let \mathbb{C}\setminus K be connected, and let f be a continuous function on K which is holomorphic and non-zero on the interior of K. Is there a sequence of polynomials \{p_n\}_{n=1}^\infty with no zeros in K such that p_n\to f?


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          Conjecture 9.2.

          [J. Andersson, D. Farmer] Let be K\subset\mathbb{C} be compact, \mathbb{C}\setminus K connected, and f be a continuous function on K which is holomorphic and non-zero on the interior of K. Suppose K\subseteq\{z:\frac{1}{2}<{\rm Re} z < 1\}, then \overline{d}\left( \left\{t\in\mathbb{R}: ||\zeta(\cdot + it) - f(\cdot)||_{\infty, K} < \epsilon\right\}\right) > 0,
          where

          \overline{d}(E):=\limsup_{T\to\infty} \frac{m(E\cap[0,T\,])}{T}.
            1. Remark. [P. Gauthier] Andersson has shown that a positive answer to Problem 9.1 and a confirmation of Conjecture 9.2 are equivalent.


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                  Problem 9.3.

                  [R. Pemantle] Let \mu be a probability measure on \{0,1\}^n. Consider f(\vec{z}\,) = \int {\vec{z}}^{\;\vec{\alpha}} d\mu(\vec{\alpha}\,)
                  and g(\vec{\lambda}\,) = \int e^{\;\vec{\lambda}\cdot\vec{\alpha}} d\mu(\vec{\alpha}\,).
                  What property of g is equivalent to the stability of f?


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                      Problem 9.4.

                      [G. Knese] Let p(z)\in\mathbb{C}[z], \deg(p)=n. Suppose p(z)\neq 0 for all |z|\le1. Define p^*(z):=z^n\overline{p\left(\frac{1}{\bar{z}}\right)}. Then it is known that

                      p(A)p(A)^\dagger \ge p^*(A)(p^*(A))^\dagger \;\; \textit{for any contractive matrix } A.


                      (A is contractive means \sup\limits_x \frac{||Ax||_2}{||x||_2}<1.) In 2D, we replace A with a pair of commuting contractions and p with a bivariate polynomial with no zeros in \{z:|z|\le 1\}\times\{w:|w|\le 1\}. The 3D generalization fails – when does it hold?


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                          Problem 9.5.

                          [P. Brändén] Find more examples and attempt to characterize Markov processes preserving stability. Explicitly, let \psi:\mathbb{R}_{MA}[x,y,z]\to\mathbb{R}_{MA}[x,y,z] be a linear operator. When is e^{t\psi}:\mathbb{R}[x,y,z]\to\mathbb{R}[x,y,z] stability preserving for all t\ge0 ?

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