Stable pencils of hyperbolic polynomials and the Cauchy problem for hyperbolic equations with a small parameter at the highest derivatives

We study pencils of hyperbolic polynomials of the form ${\mathcal R}(\tau,\xi)= \sum_{j=0}^N(-i)^j\gamma_j P_j(\tau,\xi)$, where $P_j(\tau,\xi)$is a real homogeneous polynomials of degree $m-j$ resolved with respect to the highest power of $\tau$ and $P_j(1,0)=1$; the numbers $\gamma_0,\dots ,\gamma_N$ are positive. In the first part of the paper we find necessary and close to sufficient conditions of stability of the polynomial ${\mathcal R}(\tau,\xi)$ (i.e., the condition that its roots $\tau_j(\xi)$ lie in the open upper half-plane of the complex plane). This problem is closely related to the problem on uniform (with respect to a small parameter) estimates for the solution of the Cauchy problem for hyperbolic equations with a small parameter. The latter problem (both for constant and variable coefficients) is the topic of the second part of the paper.