#### Document Type

Article

#### Publication Date

1-2003

#### Keywords

Roots of polynomials, critical points of polynomials, separation of roots

#### 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 (formula) and (formula). 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 (formula), for any polynomial of degree *n*.

#### Publication Title

Proceedings of the American Mathematical Society

#### Volume

131

#### Issue

1

#### First Page

253

#### Last Page

264

#### Required Publisher's Statement

First published in "Proceedings of the American Mathematical Society" in 2003, published by the American Mathematical Society.

#### Recommended Citation

Ćurgus, Branko and Mascioni, Vania, "On the Location of Critical Points of Polynomials" (2003). *Mathematics Faculty Publications*. 7.

https://cedar.wwu.edu/math_facpubs/7

#### Subjects - Topical (LCSH)

Polynomials; Roots, Numerical

#### Genre/Form

articles

#### Type

Text

#### Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

#### Language

English

#### Format

application/pdf

## Comments

Communicated by N. Tomczak-Jaegermann