Generalizations of parts of Grace's apolarity theorem involving circular regions (with a characteristic) and their applications

摘要

According to Grace's apolarity theorem, if the coefficient of two polynomials f(z) = ∑_(k=0)~n C(n,k)A_k z~k, g(z) = ∑_(k=0)~n C(n,k)B_k z~k, A_n B_n ≠ 0, satisfy the equation A_0 B_n-C(n,1)A_1 B_(n-1)+C(n,2) B_(n-2)+…+(-1)~n A_n B_0 = 0, then (i) f(z) has at least one zero, in a circular region C containing all zeros of g(z) (ii) g(z) has at least one zero, in a circular region C containing all zeros of f(z). We have obtained generalizations of (i), by considering g(z) to be any polynomial of degree not exceeding n and C to be a circular region (containing 0) or a circular region with a convex complement and generalizations of (ii), by considering g(z) to be any polynomial of degree not exceeding n and C to be a circular region (not containing 0) or a convex circular region. We have applied these generalizations to the study of the zeros of certain composite polynomials (obtained from two given polynomials), thereby leading also to certain generalizations of Szego's theorem [Szego, G., 1922, Bemerkungen zu einem Satz von J.H. Grace uber die Wurzeln algebraischer Gleichungen. Mathematische Zeitschrift, 13, 28-55.] involving circular regions (with a characteristic).