It is well known that an nth-order real polynomial D(z)= is Schur stable if its coefficients satisfy the monotonic condition, i.e., dn > dn-1 > > d1 > d0 > 0. In this letter it is shown that even if the monotonic condition is violated by one coefficient (say dk), D(z) is still Schur stable if the deviation of dk from dk+1 or dk-1 is not too large. More precisely we derive upper bounds for the admissible deviations of dk from dk+1 or dk-1 to ensure the Schur stability of D(z). It is also shown that the results obtained in this letter always yield the larger stability range for dk than an existing result.

In this Letter, we explore semi-definite relaxation (SDR) program for finding the real roots of a real polynomial. By utilizing the square of the polynomial, the problem is approximated using the convex optimization framework and a real root is estimated from the corresponding minimum point. When there is only one real root, the proposed SDR method will give the exact solution. In case of multiple real roots, the resultant solution can be employed as an accurate initial guess for the iterative approach to get one of the real roots. Through factorization using the obtained root, the reminding real roots can then be solved in a sequential manner.

Monotonic condition, a well-known sufficient condition for Schur stability of real polynomials, is relaxed. The condition reads that a series of strictly and monotonically decreasing positive coefficients of the polynomials yields Schur stability. It is shown by inspecting the original proof that equalities are allowed in all the inequalities but two which are located at appropriate positions.

For a real Schur polynomial, estimates are derived for a Schur stability margin in terms of matrix entries or tableau entries in some stability test methods. An average size of the zeros of the polynomial is also estimated. These estimates enable us to obtain more information than stability once a polynomial is tested to be stable via the established Schur stability criterion for real polynomials.