On critical values of polynomials with real critical points

Aimo Hinkkanen; Ilgiz Kayumov

doi:10.1007/s00365-009-9079-6

On critical values of polynomials with real critical points

Aimo Hinkkanen, Ilgiz Kayumov

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that {pipe} f(ζ)/ζ{pipe} ≤ 2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz³.

Original language	English (US)
Pages (from-to)	385-392
Number of pages	8
Journal	Constructive Approximation
Volume	32
Issue number	2
DOIs	https://doi.org/10.1007/s00365-009-9079-6
State	Published - 2010

Keywords

Critical points
Critical values
Polynomials
Smale's conjecture

ASJC Scopus subject areas

Analysis
Mathematics(all)
Computational Mathematics

Online availability

10.1007/s00365-009-9079-6

Library availability

Discover UIUC Full Text

Cite this

@article{d9a84d56583f4e05b597676822f76e26,

title = "On critical values of polynomials with real critical points",

abstract = "Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that {pipe} f(ζ)/ζ{pipe} ≤ 2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz3.",

keywords = "Critical points, Critical values, Polynomials, Smale's conjecture",

author = "Aimo Hinkkanen and Ilgiz Kayumov",

note = "Funding Information: I. Kayumov was supported by RFBR 08-01-00381. Funding Information: This material is based upon work supported by the National Science Foundation under Grant No. 0758226. This research was also supported by the grant 07365 from the Campus Research Board of the University of Illinois at Urbana–Champaign. ",

year = "2010",

doi = "10.1007/s00365-009-9079-6",

language = "English (US)",

volume = "32",

pages = "385--392",

journal = "Constructive Approximation",

issn = "0176-4276",

publisher = "Springer",

number = "2",

}

TY - JOUR

T1 - On critical values of polynomials with real critical points

AU - Hinkkanen, Aimo

AU - Kayumov, Ilgiz

N1 - Funding Information: I. Kayumov was supported by RFBR 08-01-00381. Funding Information: This material is based upon work supported by the National Science Foundation under Grant No. 0758226. This research was also supported by the grant 07365 from the Campus Research Board of the University of Illinois at Urbana–Champaign.

PY - 2010

Y1 - 2010

N2 - Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that {pipe} f(ζ)/ζ{pipe} ≤ 2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz3.

AB - Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that {pipe} f(ζ)/ζ{pipe} ≤ 2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz3.

KW - Critical points

KW - Critical values

KW - Polynomials

KW - Smale's conjecture

UR - http://www.scopus.com/inward/record.url?scp=77955575328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955575328&partnerID=8YFLogxK

U2 - 10.1007/s00365-009-9079-6

DO - 10.1007/s00365-009-9079-6

M3 - Article

AN - SCOPUS:77955575328

SN - 0176-4276

VL - 32

SP - 385

EP - 392

JO - Constructive Approximation

JF - Constructive Approximation

IS - 2

ER -

On critical values of polynomials with real critical points

Abstract

Keywords

ASJC Scopus subject areas

Online availability

Library availability

Related links

Fingerprint

Cite this