A rank theorem for Vandermonde matrices
type: Item
instance of: scholarly article
title: A rank theorem for Vandermonde matrices
zbMATH Open document ID: 1059.15002
author: Pascal Koiran
author: Natacha Portier
author: Gilles Villard
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: Let \(n\), \(r\) and \(d\) be three positive integers and \(A\) be an \(r\times (d+1)\) matrix with entries in an algebraically closed field \(K\subset {\mathbb C}\). Let \(x=(x_1, \ldots, x_n)\) be a vector of \(n\) distinct elements of \({\mathbb C}\setminus K\). Let \(V(x,A)\) be an \((n+r)\times (d+1)\) matrix such that the first \(n\) rows of \(V(x,A)\) form the \(n\times (d+1)\) Vandermonde matrix on the vector \(x=(x_1,\ldots, x_n)\) and the last \(r\) rows of \(V(x,A)\) form the matrix \(A\). If \(A\) has rank \(n\) then the matrix \(V(x,A)\) has rank \(n+r\) as soon as \(d+1\geq n(r+1)\). The authors obtain two different proofs of the above statements, the first one is based on dimension arguments and alternative proof is based on the methods from the theory of linear recurrences. The authors discuss also some extensions of the above results and its applications in the construction of the limit theory for generic polynomials.
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2081749
zbMATH Keywords: Vandermonde matrices
zbMATH Keywords: linear recurring sequences
zbMATH Keywords: model theory of fields
zbMATH Keywords: rank \(n\)
reviewed by: Alexander Guterman
MaRDI profile type: Publication
full work available at URL: j.laa.2003.09.007
OpenAlex ID: W2018708523
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DOI: 10.1016/J.LAA.2003.09.007
description: scientific article; zbMATH DE number 2081749