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\begin{document}
\title{Stability and hyperbolicity}
\author{Edited by }

\urlstub{hyperbolicpoly}

\maketitle

A list of the problems from the workshop on stability, hyperbolicity, and zero localization of functions, held December 5-9, 2011, is presented below.  The problems have been loosely categorized for convenience, although many of the problems could be listed under several different headings.


\section{Multiplier sequences and CZDS}





\begin{problemblock}

\begin{problem}[3.1] Let $\{c_k\}_{k=0}^n\subset\mathbb{R}$ be given.  Does there exist a real rooted polynomial, $p(x)$, with zeros outside $[0,n]$ such that $c_k=p(k)$, $k=0,1,\ldots,n$?  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[3.2] Let $f(z)$ be a meromorphic function.  When is the sequence $\{f(k)\}_{k=0}^\infty$ a multiplier sequence? \end{problem}




\end{problemblock}



\section{Matrix Theory}





\begin{problemblock}

\begin{problem}[.1] Study the zeros of Bessis-Moussa-Villani polynomials $t\to\text{tr}(A+tB)^m$. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Find a simple proof of BMV conjecture for $3\times 3$ matrices (and an explicit formula for the BMV measure for $3\times 3$ matrices; the existence of the measure is due to recent work of Stahl). \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Given $p\in\mathbb{C}[x]$, $\deg(p) = n$, find an $n\times n$  Hermitian matrix whose inertia $(n_+, n_{-}, n_0)$ counts the zeros of $p$ in $H_+, H_{-}, H_0=\mathbb{R}$. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Any real polynomial can be written as $\det(A-\lambda I),$ where $A$ has the (tridiagonal) form

\[  \left[\begin{array}{cccccc}
+ & + &         &          &         & \\
- & 0 & +       &          &         &\\
  & - & 0       & +        &         &\\  
  &   & \ddots  & \ddots   & \ddots  &  \\
  &   &         &     -    &   0     &  + \\  
  &   &         &          &   -     &  -   \\ 
\end{array}\right].\]
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.5]  Investigate the signs of Hurwitz minors for P\'olya frequency sequences. \end{problem}




\end{problemblock}



\section{Log Concavity}

Let $\mathcal{L}\text{-}\mathcal{P}$ denote the class of functions which are locally uniform limits of polynomials having only real zeros, and let the class of functions in $\mathcal{L}\text{-}\mathcal{P}$ which have only non-negative Maclaurin coefficients be denoted $\mathcal{L}\text{-}\mathcal{P}^+$.  



\begin{problemblock}

\begin{problem}[.1] Let 
\[Q_n^{(\alpha, \beta)}:=\sum_{k=0}^n\binom{n}{k}f_kf_{n-k}((x+\alpha)_k(x+\beta)_{n-k} - (x+\alpha+\beta)_k(x)_{n-k}),\]
where $\alpha, \beta>0$, $f_k^2\ge f_{k-1}f_{k+1}$, and $(x)_k = x(x+1)\cdots(x+k-1)$.  Then the Maclaurin coefficients of $Q^{(\,\alpha,\,\beta)}$ are non-negative. \end{problem}




\begin{remark}
\by{D. Karp}I have proved the case $\alpha=\beta=1$ and formulated a number of generalizations of this conjecture in http://arxiv.org/abs/1203.1482\end{remark}

\end{problemblock}



\begin{problemblock}The following conjecture of (McNamara et al.) was proved by P. Br\"and\'en.  Let $p(x):=\sum_{k=0}^n a_k  x^k$, with $a_{-1}:=a_{n+1}:=0$.
If $p\in\mathcal{L}\text{-}\mathcal{P}^+$, then 
\[\sum_{k=0}^n  \left|\begin{array}{cc}
a_k & a_{k+1} \\
a_{k-1} & a_{k} \end{array}\right|x^k \in\mathcal{L}\text{-}\mathcal{P}.\]
S. Fisk generalized this statement to polynomials with coefficients formed from the determinants of $3\times 3$ matrices in the conjecture below.

\begin{conjecture}[.2] \label{fidk_conj}
If $p\in\mathcal{L}\text{-}\mathcal{P}^+$, then
\[\sum_{k=0}^n  \left|\begin{array}{ccc}
a_k & a_{k+1} & a_{k+2} \\
a_{k-1} & a_k & a_{k+1}\\
a_{k-2} & a_{k-1} & a_k\end{array}\right|x^k\in\mathcal{L}\text{-}\mathcal{P}^+.\] \end{conjecture}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Let $\{\phi_k(x)\}_{k=0}^\infty$ and $\{f_k\}_{k=0}^\infty$ be log-concave.  When is $\sum f_k\phi_k(x)$ log-concave ? \end{problem}




\end{problemblock}



\section{Combinatorics and Graph Theory}





\begin{problemblock}Let $S_{2n}$ be the convex hull of all adjacency matrices of perfect matchings.  Let
\[\mu_{2n}:=\min\{ \text{haf}(B):B\in S_{2n}\}, \]
where
\[\text{haf}(B):=\frac{1}{n!2^n}\frac{\partial}{\partial x_1\cdots\partial x_{2n}}(x^{T}Bx)^n \]
is the \emph{hafnian} of $B$.

\begin{problem}[.1] \begin{enumerate}
\item Is it true that
\[\alpha:=\liminf_n \frac{\log\mu_{2n}}{2n}>-\infty \text{ ?}\]

\item Estimate $\mu_{2n}$.

\end{enumerate} \end{problem}




\end{problemblock}



\begin{problemblock}The type-$D_n$ Eulerian polynomials are given by
\[ E(x) : = \sum_w x^{\#\{s:\text{length}(ws)<\text{length}(w)\}}. \]

\begin{problem}[.2] 
Prove that E(x) has only real zeros. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Is there a decent multivariate extension of the Chudnovsky-Seymour Theorem that the vertex independent set polynomial of a claw-free graph has only real roots? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Generalize the Chudnovsky-Seymour (Heilmann-Lieb) Theorem.  Extend the common interlacing property to several variables. \end{problem}




\end{problemblock}



\section{Zeros of derivatives}





\begin{problemblock}

\begin{problem}[.1] Let $p\in\mathbb{R}[x]$, $\deg(p)=n>2$.  There are at least $n-1$ points of extreme curvature $\kappa(p)$ ( more generally where $\kappa'(p)=0$ ) \cite{MR2104693}. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] What are the zero spacing effects of higher order differential operators on functions which belong to $\mathcal{L}\text{-}\mathcal{P}$? \end{problem}




\end{problemblock}



\section{Orthogonal polynomials}





\begin{problemblock}Let $(L_n^\alpha)$ denote the generalized Laguerre polynomials.  It is known that $L_m^\alpha$ and $L_n^\alpha$ can have common zeros if $|m-n|>1$.  Is the number of common zeros linked to arithmetic properties of $\alpha$?

\begin{conjecture}[.1]  There are no common zeros between $L_m^\alpha$ and $L_n^\alpha$, $|m-n|>1$, for $\alpha$ rational. \end{conjecture}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Let $P_n^{\;\alpha, \,\beta}$ denote the Jacobi polynomial of $\deg n$ with parameters $\alpha, \beta> -1$.  
Then $P_n^{\;\alpha', \,\beta'}$ and $P_n^{\;\alpha, \,\beta}$ interlace for $\alpha\approx\alpha'$, and $\beta\approx\beta'$.  Does the failure of interlacing correspond to some physical law being ``violated''? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Consider a subspace $S$ of $\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given).  Given a scalar product on $\mathbb{C}[x]$, find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.
\begin{enumerate}
\item Which degrees get skipped?
\item Characterize these orthogonal polynomials.
\item Does this have anything to do with lacunary Pad\'e approximants?
\end{enumerate}
 \end{problem}




\end{problemblock}



\section{Riemann $\Xi$-function}

Let 
\[\Xi(z):=\int_{-\infty}^{\infty} \Phi(t)\cos(zt) dt,
\]
where
\[\Phi(t):=\sum_{n=1}^\infty (2n^4\pi^2e^{9t}-3n^2\pi e^{5t})\exp(-n^2\pi e^{4t}).\]



\begin{problemblock}
Let $p_k(z)$ be the polynomials orthogonal with respect to $\Phi$ on $(-\infty,\infty)$. 
Define
\[P_n(z):=\frac{\det\left[p_i(z_j)\right]_{i,j=1}^n}{\det\left[z_i^{j-1}\right]_{i=1,j=1}^n}, \qquad\text{and}\qquad
Q_n(z):=\frac{\det\left[p_i(z_j)\right]_{i,j=2}^n}{\det\left[z_{i}^{j-2}\right]_{i,j=2}^{n}}.\] 

\begin{problem}[.1] Investigate the zeros of $P_n$ and $Q_n.$
 \end{problem}




\begin{remark}
\by{D. Dimitrov}The following generalization of the result mentioned above is true.

Let $\mu$ be an even, positive measure on $\mathbb{R}$, and $F(z):=\int_{-\infty}^\infty\cos(zt)d\mu(t)$.  Let $\{p_k(z)\}_{k=0}^\infty$ be the orthogonal polynomials with respect to $\mu$ on $(-\infty,\infty)$.  Then $F\in\mathcal{L}\text{-}\mathcal{P}$ if and only if the associated polynomial $P_n[p_1,\ldots,p_n]$ has purely imaginary zeros, if and only if $Q_n[p_1,\ldots, p_n]$ has no purely imaginary zeros. (for all $n$)\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[8.2] Are there any non-real zeros of 
\[\int_{0}^\infty \Phi^\alpha(t)\cos(zt)dt \qquad \text{for}\qquad \alpha>0 ?\] \end{problem}




\end{problemblock}



\section{Zeros of specific functions}





\begin{problemblock}

\begin{conjecture}[.1]  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 $n$. \end{conjecture}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] 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 $\mathcal{L}\text{-}\mathcal{P}$? \end{problem}




\end{problemblock}



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


\begin{conjecture}[.3] For all $|w|>1$, all the zeros of $P_n(\cdot,w)$ are simple and separated in modulus (by a factor of at least $|w|^2$). \label{conj_sokal} \end{conjecture}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Generalize conjecture \ref{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? \end{problem}




\end{problemblock}



\section{Miscellaneous Problems}





\begin{problemblock}

\begin{problem}[.1] \label{nonzero_approx}
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 $p_n\to f$? 
 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{conjecture}[.2] \label{conj_universal}
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

\[\overline{d}(E):=\limsup_{T\to\infty} \frac{m(E\cap[0,T\,])}{T}.\]
 \end{conjecture}




\begin{remark}
\by{P. Gauthier}Andersson has shown that a positive answer to
Problem \ref{nonzero_approx} and a confirmation of Conjecture \ref{conj_universal} are equivalent.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] 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 $f$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] 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 \emph{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? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.5] 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 $t\ge0$ ? \end{problem}




\end{problemblock}





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\end{document}
