Detection of positive roots of a polynomial with five parameters (Q5953948)

The authors consider the polynomial NEWLINE\[NEWLINE f(\lambda) = \lambda^{\tau+\sigma}(\lambda-1)^n + p\lambda^\sigma(\lambda-1)^n + q\lambda^\tau NEWLINE\]NEWLINE with the five parameters \(n\), \(\tau \in \mathbb N\), \(\sigma \in \mathbb N_0\) and \(p\), \(q \in \mathbb R\). This polynomial arises from the characteristic equation of the neutral difference equation with delay NEWLINE\[NEWLINE \Delta^n(x_k+px_{k-\tau})+qx_{k-\sigma}=0 , \quad k \in \mathbb N_0 , NEWLINE\]NEWLINE which is used to model population dynamics. The authors give a complete set of discriminatory criteria for \(f\) to have a positive root. Their investigations are based on the theory of envelopes.