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## 2. Matrix Theory

1. #### Problem 2.1.

[O. Holtz] Study the zeros of Bessis-Moussa-Villani polynomials $t\to\text{tr}(A+tB)^m$.
• #### Problem 2.2.

[P. Moussa] 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).
• #### Problem 2.3.

[A. Sokal] 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 $H_+, H_{-}, H_0=\mathbb{R}$.
• #### Problem 2.4.

[M. Tyaglov] Any real polynomial can be written as $\det(A-\lambda I),$ where $A$ has the (tridiagonal) form

$\left[\begin{array}{cccccc} + & + & & & & \\ - & 0 & + & & &\\ & - & 0 & + & &\\ & & \ddots & \ddots & \ddots & \\ & & & - & 0 & + \\ & & & & - & - \\ \end{array}\right].$
• #### Problem 2.5.

[L. Grabarek, M. Tyaglov] Investigate the signs of Hurwitz minors for Pólya frequency sequences.

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