Real Wronskian zero of polynomials with nonreal zeros
type: Item
instance of: scholarly article
title: Real Wronskian zero of polynomials with nonreal zeros
zbMATH Open document ID: 0715.30008
DOI: 10.1016/0022-247X(91)90078-E
author: Karl Dilcher
published in: Journal of Mathematical Analysis and Applications
publication date: 1991-01-01T00:00:00Z
review text: Let f be a polynomial with real coefficients and let \[ Wf(z)=f(z)f''(z)- (f'(z))^ 2 \] be the ``Wronskian'' of f. This paper is devoted to studying the relationship between the geometry of the zeros of f and that of Wf. The results are aimed at a conjecture of Craven, Csordas and Smith which states that the number of real zeros of Wf never exceeds that of f. The author's main results, as summarized in his introduction, are as follows (recall that the Jensen disk of a pair of conjugate complex numbers is the closed disk whose diameter is the line segment joining these two numbers): 1. Any nontrivial real zero of Wf lies in the Jensen disk of some pair of complex conjugate zeros of f. (A trivial real zero of Wf is one which is a multiple zero of f). 2. If a pair of complex conjugate zeros of f is sufficiently isolated from other zeros of f, then the associated closed Jensen disk contains exactly two real or complex zeros of Wf. 3. If there are real zeros of f sufficiently close to a pair of complex zeros of f, then the Jensen disk associated with these complex zeros contains no real zeros of Wf. 4. If, in a fixed interval, f has sufficiently many real zeros (depending on the number and location of the complex zeros of f), then Wf has no real zeros. 5. If the zeros of f lie in two circles symmetric to and sufficiently distant from the real axis, then Wf has exactly two real zeros. 6. If f has no multiple zeros, then the complex zeros of Wf outside the Jensen disks of f cannot lie too close to the real axis.
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 4177462
zbMATH Keywords: Wronskian
reviewed by: David W. Boyd
MaRDI profile type: Publication
cites work: Verallgemeinerung eines Lemmas von J. L. Walsh nebst einigen Anwendungen
cites work: Erratum to "On the Location of the Zeros of the Derivative of a Polynomial"
cites work: The zeros of derivatives of entire functions and the Pólya-Wiman conjecture
cites work: Zeros of the Wronskian of a polynomial
cites work: A Generalization of a Theorem of Bôcher
cites work: Q5800921
full work available at URL: 0022-247x(91)90078-e
OpenAlex ID: W2071735806
description: scientific article; zbMATH DE number 4177462