Article

On the Location of Critical Points of Polynomials

Proceedings of the American Mathematical Society
(2003)
Abstract

Given a polynomial

**of degree***p***and with at least two distinct roots let***n ≥ 2***. For a fixed root***Z(p) = {z : p(z) = 0}***we define the quantities***α ∈ Z(p)***and***ω(p,α) := min{|α - v| : v ∈ Z(p) \ {α}}***. We also define***τ(p, α) := min{|α - v| : v ∈ Z(p') \ {α}}***and***ω(p)***to be the corresponding minima of***τ(p)***and***ω(p,α)***as***τ(p,α)***runs over***α***. Our main results show that the ratios***Z(p)***and***τ(p,α)/ω(p,α)***are bounded above and below by constants that only depend on the degree of***τ(p)/ω(p)***. In particular, we prove that***p***, for any polynomial of degree***(1/n)ω(p) ≤ τ(p) ≤ (1/2 sin(π/n))ω(p)***.***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/