By Jean-Pierre Aubin
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Implies that every zero z of (3) satisfies an inequality J z - (ao + Au") I < e, where A11" is a suitably chosen n-th root of A. By means of Theorem 1 we shall establish the COROLLARY. When. z becomes infinite, a zero a of the equation in a (4) p(z) = (z - a)" approaches the point ao . That is to say, let e > 0 be given; there exists R. such that I z I > R. implies that equation (4) has a zero a in the region I a - ao I < e. Of course I p(z) I becomes infinite when and only when I z I becomes infinite, so the Corollary is a consequence of Theorem 1.
REAL POLYNOMIALS of the one polynomial are found by that same transformation from the critical points of the other. If we have X < 1, all the points a; , b; , and the intervals Ix lie on one side of y, in the order indicated by their subscripts, and the intervals I,, of Corollary 3 are precisely the intervals I,. of Theorem 1; Corollary 3 follows from Theorem 1. -1 (notation of Corollary 3) according as we have y => a or y < a1 contains all critical points of p(z); Corollary 3 is a consequence of Theorem 1.
POLYNOMIALS WITH REAL ZEROS 25 the points a; to the right one at a time, while they trace their respective loci. The k-th zero of p'(z) increases continuously and monotonically during this process, starting from the point ck , and reaching but never going beyond the point dk . 2). Several special cases of this theorem are of interest. COROLLARY 1. [for k = 1, Nagy, 1918; in part due to Laguerre]. Let a, be algebraically the least zero of the polynomial p(z) with only real zeros; denote the multiplicity of a, by k and the degree of p(z) by n.