Location analysis of complex trinomial roots with respect to the unit circle in the complex plane

Stability analysis of linear difference equation with constant coefficients is closely related to a location of roots of its characteristic polynomial, particularly with respect to the unit disk in the complex plane. We focus to a particular case of a trinomial with two strictly complex coefficients \begin{equation*} T_{k,m}(\lambda)=\lambda^k+\mathrm{i}a\lambda^{k-m}+\mathrm{i}b, \end{equation*} where $a,b\in\mathbb{R}$ and $k>m$, $k,m\in\mathbb{N}$. Our aim is to analyze a relation between number of roots of the trinomial $T_{k,m}$ with a modulus lower than one, equal to one and greater than one, and a pair of parameters $(a,b)$. The root locus technique is utilized to obtain regions in the $(a,b)$ plane, where a number of roots of $T_{k,m}$ with modulus lower than one is preserved. Several figures will be introduced to demonstrate the above mentioned regions for particular cases of $T_{k,m}$ together with relevant remarks about their properties. It is also possible to use an introduced procedure in another special cases of polynomials. The talk is based on a joint work with Ji\v{r}\'\i\, J\'{a}nsk\'{y}.