The Hankel pencil conjecture
type: Item
instance of: scholarly article
title: The Hankel pencil conjecture
zbMATH Open document ID: 1176.15038
author: Alexander Kovačec
author: M. C. Gouveia
published in: Linear Algebra and its Applications
publication date: 2009-09-14T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/11297
review text: For \(n\geq3\) consider the Hankel \(n\times n\) matrix \(H_{n}(x)=H_{n} (x;c_{1},\dots ,c_{n+1})\) over \(C[x]\) where the \((i,j)\)th entry is equal to \(x\) if \(i+j=n-1\); to \(c_{i+j-n+1}\) if \(i+j\geq n\); and to \(0\) otherwise. The ``Hankel pencil conjecture'' (HPnC) states that if \(H_{n}(x)=0\) and \(c_{1},\dots ,c_{n+1}\) are nonzero complex numbers, then the last two columns of \(H_{n}(x)\) are linearly dependent. This conjecture was proved for \(n\leq5\) by \textit{W. Schmale} and \textit{P. K. Scharma} [Linear Algebra Appl. 389, 33--42 (2004; Zbl 1068.15008)], but a purported proof by V. Bolotnikov and C. K. Li of the general case has been shown to have a significant gap. The authors of the present paper show that for degree \(n\) HPnC is implied by a new conjecture RnC (the ``root conjecture'') concerning the roots of certain polynomials of degree \(n-2\). Computations are used to verify RnC (and hence HPnC) for all \(n\leq8\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 5603569
zbMATH Keywords: Hankel matrices
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: Sylvester identity
zbMATH Keywords: Hankel pencil conjecture
zbMATH Keywords: root conjecture
zbMATH Keywords: systems of polynomial equations
MaRDI profile type: Publication
OpenAlex ID: W2136092747
Wikidata QID: Q123010625
cites work: Q3050437
cites work: Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley
cites work: Quasidirect decompositions of Hankel and Toeplitz matrices
cites work: Matrix Analysis
cites work: Cyclizable matrix pairs over \(\mathbb C[x]\) and a conjecture on Toeplitz pencils
DOI: 10.1016/J.LAA.2009.05.015
description: scientific article; zbMATH DE number 5603569