On the boundary of the set of Schur polynomials and applications to the stability of 1-D and 2-D digital recursive filters

The authors show that implementation of stability test for 2-D digital quarter-plane or nonsymmetric half-plane recursive filters requires the testing of whether a particular resultant vanishes on the unit circle. The authors prove that this cannot be avoided, whatever the nature of implementation may be for the stability test. This result is established by studying the set of all the Schur polynomials whose coefficients belong to the space of univariate complex polynomials of degree not greater than n. The authors first prove that this set of Schur polynomials is connected. Next, they give the equation, which is obtained by equating a particular resultant to zero, of the smallest hypersurface containing the boundary of this set. Finally, it is shown that this equation is irreducible