Further results on the bounds of the zeros of quasi-critical polynomials

On the basis of the relationship of the mth power of a polynomial and its modular form (polynomial whose coefficients are the moduli of the coefficients of that polynomial), we derive a necessary and sufficient condition for the modulus of the mth power of a polynomial for contacting its modular form on the boundary of a disc. Combined with the result about distribution of zeros of analytic function, some new sufficient conditions are derived which give bounds of the absolute values of the roots of a quasi-critical polynomial. These results extend certain earlier similar tests for linear discrete-time systems. Finally, four examples are given to demonstrate the results, Example 2.1 gives a state feedback application, Examples 2.2 and 2.4 deal with r-stability, and Example 2.3 display that our theorems give better results when m increases but at the cost of increasing complexity.