The robustness properties of univariate and multivariate reciprocal polynomials

The author investigates the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit circle and the unit polycircle, respectively. It is shown that any nonseparable collection of univariate reciprocal polynomials is nonzero on the unit circle, if and only if a set of real-valued rationals corresponding to the vertices of the convex hull of that collection are entirely positive or negative on the unit circle. It is shown that this result generalizes to the case of multivariable polynomials: for a nonseparable collection of multivariate polynomials to be nonzero on the unit polycircle, it is necessary and sufficient that a set of real-valued multivariate rationals corresponding to the vertices of the convex-hull of that collection are entirely either positive or negative on the unit polycircle. The applicability of this result in signal and image processing, spectrum estimation, and stochastic modeling is discussed, and some examples of its usefulness are given