Please use this identifier to cite or link to this item:
Title: The Hankel Pencil Conjecture
Authors: Kovačec, Alexander 
Gouveia, Maria Celeste 
Keywords: Hankel matrices; Toeplitz matrices; Systems of polynomial equations; Sylvester identity
Issue Date: 2007
Publisher: Centro de Matemática da Universidade de Coimbra
Citation: Pré-Publicações DMUC. 07-17 (2007)
Abstract: The Toeplitz pencil conjecture stated in [SS1] and [SS2] is equivalent to a conjecture for n £ n Hankel pencils of the form Hn(x) = (ci+j¡n+1); where c0 = x is an indeterminate, cl = 0 for l < 0; and cl 2 C¤ = Cn f0g; for l ¸ 1: In this paper it is shown to be implied by another conjecture, we call root conjecture. This latter claims for a certain pair (mnn;mn¡1;n) of submaximal minors of certain special Hn(x) that, viewed as elements of C[x]; there holds that roots(mnn) µ roots(mn¡1;n) implies roots(mn¡1;n) = f1g: We give explicit formulae in the ci for these minors and show the root conjecture for minors mnn;mn¡1;n of degree · 6: This implies the Hankel Pencil conjecture for matrices up to size 8 £ 8: Main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices.
Rights: openAccess
Appears in Collections:FCTUC Matemática - Vários

Files in This Item:
File Description SizeFormat
The Hankel Pencil Conjecture.pdf238.3 kBAdobe PDFView/Open
Show full item record

Page view(s) 50

checked on Aug 4, 2022

Download(s) 50

checked on Aug 4, 2022

Google ScholarTM


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.