Constrained polynomial roots and a modulated approach to Schur stability

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the characteristic polynomial is more than quartic. In this paper, we reverse the classical process and use the authors' work on global stability to find sufficient conditions on the coefficients that ensure the zeros of the characteristic polynomial are in the open unit disk. This leads to an algorithm that begins by testing the $\ell_1$-norm of the polynomial, and if it is not less than two, perform an iteration process that can be implemented with moderate effort. We give examples that show the effectiveness of our method when compared with the Jury's algorithm. Last, we formalize our constructions in terms of semialgebraic sets.