Clearly has degree at most . Then by a theorem of Schur is a Schur polynomial if and only if and is a Schur polynomial.

Clearly, the interval is an interval of absolute stability if, for all the (real) stability polynomial , is a Schur polynomial. Writing for , we can construct and . The first condition yields our first inequality in , while the second condition may be tested by writing for and repeating the process, there by obtaining a second inequality for , and so on. At each stage, the degree of the polynomial under test in reduced by one, so that eventually we merely have to state a criterion for a polynomial of degree one to be a Schur polynomial, and, obviously, this can easily be done.

We cannot use the Schur criterion directly to determine intervals of relative stability is the sense previously defined. However, if we adopt a definition of relative stability which requires that , then it is technically possible to use the Schur criterion. Substituting into (8.34) gives a polynomial equation in

If the roots of this equation are , then the Schur criterion gives necessary and sufficient conditions for , that is, for