The robustness properties of univariate and multivariate reciprocal polynomials

Abstract

The author investigates the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit-circle and the unit-polycircle, respectively. He shows 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 described by Lakshmanan (1992). 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. He shows 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 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 he develops the results for polytopic families, he then extends those results to the case of non-polytopic reciprocal polynomial families. >