On the Lorentz degree of a product of polynomials
type: Item
instance of: scholarly article
title: On the Lorentz degree of a product of polynomials
zbMATH Open document ID: 1305.41005
author: Rachid Ait-Haddou
published in: Journal of Approximation Theory
publication date: 2014-11-28T00:00:00Z
review text: The smallest integer \(m\) for which a real polynomial \(p\) having no zeros in an interval \((a,b)\) admits a representation of the form \(p(x) = \sum_{k=0}^{m} c_k (x-a)^k (b-x)^{m-k}\) with all \(c_k \geq 0\) or all \(c_k \leq 0,\) is called Lorentz degree of \(p\) over \([a,b]\) and denoted by \(d_{[a,b]}(p)\). \textit{T. Erdélyi} [J. Approx. Theory 67, No. 2, 187--198 (1991; Zbl 0759.41003)] formulated the following two conjectures: Is it true that \(d_{[a,b]}(pq) \geq \min (d_{[a,b]}(p), d_{[a,b]}(q))\) for any polynomials \(p\) and \(q\)? Is it true that \(d_{[a,b]}(pq) \geq | d_{[a,b]}(p) - d_{[a,b]}(q) |\) for any polynomials \(p\) and \(q\)? The author answers negatively the above two questions. The validity of the first one was shown for exact degree two and for a family of other high degree polynomials.
reviewed by: Zoltán Finta
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 6374936
zbMATH Keywords: polynomials with nonnegative coefficients
zbMATH Keywords: Bernstein bases
zbMATH Keywords: Lorentz degree
zbMATH Keywords: Bézier curves
zbMATH Keywords: degree elevation
zbMATH Keywords: Pólya degree
MaRDI profile type: Publication
full work available at URL: j.jat.2014.10.001
OpenAlex ID: W2036775107
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DOI: 10.1016/J.JAT.2014.10.001
description: scientific article; zbMATH DE number 6374936