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Article
On the Location of Critical Points of Polynomials
Proceedings of the American Mathematical Society (2003)
  • Branko Ćurgus, Western Washington University
  • Vania Mascioni
Abstract
Given a polynomial p of degree n ≥ 2 and with at least two distinct roots let Z(p) = {z : p(z) = 0}. For a fixed root α ∈ Z(p) we define the quantities ω(p,α) := min{|α - v| : v ∈ Z(p) \ {α}}and τ(p, α) := min{|α - v| : v ∈ Z(p') \ {α}}. We also define ω(p) and τ(p) to be the corresponding minima of ω(p,α) and τ(p,α) as α runs over Z(p). Our main results show that the ratios τ(p,α)/ω(p,α) and τ(p)/ω(p) are bounded above and below by constants that only depend on the degree of p. In particular, we prove that (1/n)ω(p) ≤ τ(p) ≤ (1/2 sin(π/n))ω(p), for any polynomial of degree n.
Keywords
  • Roots of polynomials,
  • Critical points of polynomials,
  • Separation of roots
Disciplines
Publication Date
2003
Publisher Statement
© Copyright 2015, American Mathematical Society DOI: http://dx.doi.org/10.1090/S0002-9939-02-06534-6
Citation Information
Branko Ćurgus and Vania Mascioni. "On the Location of Critical Points of Polynomials" Proceedings of the American Mathematical Society Vol. 131 Iss. 1 (2003)
Available at: http://works.bepress.com/branko_curgus/34/