Some Remarks on the -stability for Families of Polynomials

Using Brouwer degree, we prove a more general version of the zero exclusion principle for families of polynomials and apply it to obtain very simple proofs of extensions of recent results on the Routh-Hurwitz and Schur-Cohn stability of families of polynomials. 1. Introduction It is well know that the problem of the stability of a linear diierence or differential system with constant coeecients reduces to the question of locating the roots of its characteristic polynomial in a suitable region of the complex plane. The Routh-Hurwitz and Schur-Cohn tests, which respectively correspond to the open left half-space and the open unit ball, are well known in this respect 2, 3]. In a recent work 9], Zahreddine has considered the following problem: given a path-wise connected region in the complex plane and a set S of polynomials of the same degree, nd conditions under which all polynomials of S have their zeros inside : In the special case where S is made of all the convex combinations of two polynomials which are stable in the Routh-Hurwitz or the Schur-Cohn sense, Zahreddine has found necessary and suucient conditions for this set S to have the same stability. His approach is algebraic and based upon some properties of resultants and standard Routh-Hurwitz or Schur-Cohn stability conditions for a complex polynomial 2, 3]. The aim of this note is to show that a very simple proof of more general version of this result can be obtained by using the elementary properties of the Brouwer degree 4]. We rst use this technique to prove a more general version of the standard zero exclusion principle 1, 6] and then apply it to the proof of the Zahreddine's results.