A family of new Rudin´s theorem type results for scattering Schur polynomials

The author proves a class of results which can be considered to be the scattering Schur counterpart of the Rudin type results well known in the context of strict sense Schur polynomials. These results establish that test for zeros of a multivariable polynomial in a polydisc can be equivalently broken up into a set of simpler tests, the latter tests being a search for zeros in domains which are subsets of the unit polydisc. The author classifies the results of the present work into two categories. Results of the first type require test for zeros of the polynomial on the distinguished boundary and a set of 1-D tests for zeros in the unit disc. Results of the second type require a test for zeros of the polynomial on the distinguished boundary and only one 1-D test in the unit disc