By Morris Marden
Throughout the years because the first variation of this recognized monograph seemed, the topic (the geometry of the zeros of a posh polynomial) has persisted to reveal an identical striking power because it did within the first a hundred and fifty years of its historical past, starting with the contributions of Cauchy and Gauss. hence, the variety of entries within the bibliography of this variation needed to be elevated from approximately three hundred to approximately six hundred and the ebook enlarged by way of one 3rd. It now contains a extra huge therapy of Hurwitz polynomials and different subject matters. the recent fabric on infrapolynomials, summary polynomials, and matrix equipment is of specific curiosity.
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Additional info for Geometry of polynomials, Second Edition
Let N = IT m; - I1 n, and co(z) _ ji (z - a;) j1(z - f;). Then every finite zero of R'(z), not an a; , is a zero for which of the extremal polynomial of form p(z) = Nzm+n-1 + bizm+n-2 + p(a;) = m;co'(a;) (j = 1,2, , m) and the Tchebycheff norm max l p(#;)ln;co'(#;)l , fln} [Shisha-Walsh 3]. Hint: Cf. ex. (5,6). is a minimum on the set Ai CHAPTER II THE CRITICAL POINTS OF A POLYNOMIAL 6. The convex hull of critical points. In the previous chapter, we found that any critical point (zero of the derivative) of the polynomial (6,1) f(Z) = (Z - Zi)ml(Z - Z2)m2 .
Let G be the Green's function with pole at infinity, for an infinite region R having as boundary a finite set B of Jordan curves, which are symmetric in the real axis. Then every non-real critical point of G lies in or on at least one of the circles whose diameters join the pairs of symmetric points of B. EXERCISES. Prove the following. 1. If a is a real constant and if f(z) is a real polynomial whose derivative is f'(z), none of the imaginary zeros of F1(z) = (D + a) f (z) =f(z) + of (z) lies outside the Jensen circles off (z).
We may also state results analogous to Lucas' Theorem (6,1) for three real variables [see ex. (6,10)], for a quaternion variable [see Scheelbeek 1] and for an abstract field [see Zervos 3]. Finally, we may state the following theorem due to Walsh [20, p. 249] regarding the critical points of Green's function. THEOREM (6,5). Let R be an infinite region whose boundary B consists of a finite set of Jordan curves, and G the Green's function for R with pole at infinity. Then all the critical points of G in R lie in the convex hull H of B; none lies on the boundary of H unless the points of B are collinear.