THE ROBUSTNESS PROPERTIES OF UNIVARIATE AND MULTIVARIATE RECIPROCAL POLYNOMIALS

In this paper, we investigate the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit-circle and the unit-polycircle, respectively. We show that any polytope of univariate reciprocal polynomials are nonzero on the unit-circle, if and only if a set of real-valued rationals corresponding to its vertices are entirely either positive or negative on the unit-circle. Ensuring that these vertex rationals are entirely either positive or negative on the unit-circle can be carried out by the tests in [13] and [15]. When these existing tests are combined with the results contained in this paper, it provides a complete procedure for testing the nonzeroness of polytopes of univariate reciprocal polynomials over the unit-circle. Next, we show that this result generalizes to the case of multivariate polynomials: For any polytope of multivariate polynomials to be nonzero on the unit-polycircle, it is necessary and sufficient that a set of real-valued multivariate rationals corresponding in [14] and [15] to its vertices are entirely either positive or negative on the unit-polycircle. Again, by using the test, the positivity or the negativity of the vertex rationals can be ensured as well, thereby resulting in a complete procedure for testing the nonzeroness of an entire polytope of multivariate reciprocal polynomials over the unit-polycircle. Although we develop our results for polytopic families, we then extend those results to the case of non-polytopic reciprocal polynomial families. We show that any non-separable collection of univariate (or multivariate) reciprocal polynomials is nonzero over the unit-circle (or -polycircle) if and only if a set of real-valued rationals corresponding to the vertices of the convex-hull of that collection are entirely either positive or negative on the unit-circle (or -polycircle). Finally, we provide a couple of examples that illustrate the usefulness of these results.