On the critical points of a polynomial

Abstract

Let p be a complex polynomial, of the form , where |zk| ≥ 1 when 1 ≤ k ≤ n − 1. Then p′(z) ≠ 0 if |z| /n.Let B(z, r) denote the open ball in with centre z and radius r, and denote its closure. The Gauss-Lucas theorem states that every critical point of a complex polynomial p of degree at least 2 lies in the convex hull of its zeros. This theorem has been further investigated and developed. B. Sendov conjectured that, if all the zeros of p lie in then, for any zero ζ of p, the disc contains at least one zero of p′; see [3, Problem 4.1]. This conjecture has attracted much attention-see, for example, [1], and the papers cited there. In connection with this conjecture, Brown [2] posed the following problem.

On Zero Free Regions for Derivatives of a Polynomial

“…He also conjectured in the same paper that C n = 1 n . This problem was solved by Aziz and Zarger [1]. In this paper, we obtain the results which generalizes the results of Aziz and Zarger.…”

“…However, Brown himself conjectured that C n = 1 n . This problem has been settled by Aziz and Zarger [1], in fact they proved the following.…”

Cognition in the formal modes: Research mathematics and the SOLO taxonomy

“…For instance, I received a referee's report detailing a shorter way of illustrating an example I had discovered, but the referee acknowledged that "I am unlikely to have found it [the example] if I had not seen your work" (Anonymous referee, personal communication, January 1998). Mathematical papers are often graced with footnotes thanking referees for helping to simplify the proof of a result (e.g., Aziz & Zargar, 1998), which, of course, means that the structure of the observed learning outcome is changed before publication, usually to something less complex. Note, however, that the original prover still retains authorship and credit for the result, since from the viewpoint of the discipline it is the knowledge which is important and not the structure.…”

On zero-free regions for the derivative of a polynomial

“…Taking m = 2 we get For the proofs of these theorems we need the following result which is due to Aziz and Zagar [5]. …”