Vol. 103, No. 1, 1982

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On the zeros of composite polynomials

Abdul Aziz

Vol. 103 (1982), No. 1, 1–7

Let P(z) = Σj=0nC(n,j)Ajzj and Q(z) = Σj=0nC(n,j)Bjzj, AnBn0, be two polynomials of the same degree n. If P(z) and Q(z) are apolar and if one of them has all its zeros in a circular region C, then according to a famous result known as Grace’s theorem, the other will have at least one zero in C. In this paper we propose to relax the condition that P(z) and Q(z) are of the same degree. Instead, we will assume P(z) and Q(z) to be the polynomials of arbitrary degree n and m respectively, m n, with their coefficients satisfying an apolar type relation and obtain certain generalizations of Grace’s theorem for the case when the circular region C is a circle |z| = r. As an application of these results, we also generalize some results of Szegő, Cohn and Egerváry.

Mathematical Subject Classification 2000
Primary: 30C15
Secondary: 26C10
Received: 3 April 1981
Published: 1 November 1982
Abdul Aziz