To determine that all the roots of a transcendental polynomial have negative real parts (Q1364969)

The paper deals with polynomials of the type \(H(z,t)=\sum_{m,n}a_{m,n}z^mt^n,\) where \(a_{m,n},z,t\in\mathbb{C},\) and \(m,n\in\mathbb{N}_0\). Denote by \(\Omega_\tau,\) \(\tau\geq0,\) the number of \(p\)-roots (i.e. roots with positive real part) of \(H(z,t)\). Under some assumptions for \(H(z,t)\) (one of them is that \(H\) possesses a principle term) a method is given to determine \(\Omega_0\). A polynomial is said to be stable if all of his roots have negative real parts. The stability of certain polynomials is investigated. As application the stability of some differential-difference equations is investigated.