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Abstract. We study pencils of hyperbolic polynomials of the form R (τ, ξ, where P j (τ, ξ) is a real homogeneous polynomials of degree m − j resolved with respect to the highest power of τ and P j (1, 0) = 1; the numbers γ 0 , . . . , γ N are positive. In the first part of the paper we find necessary and close to sufficient conditions of stability of the polynomial R(τ, ξ) (i.e., the condition that its roots τ j (ξ) 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.