A generalization of Descartes rule of signs and fundamental theorem of algebra
type: Item
instance of: scholarly article
title: A generalization of Descartes rule of signs and fundamental theorem of algebra
zbMATH Open document ID: 1247.12002
author: Pentti Haukkanen
author: Timo Tossavainen
published in: Applied Mathematics and Computation
publication date: 2011-11-22T00:00:00Z
review text: In this paper there are described axiomatically families of real functions defined on some interval and having the property that each linear combination of such functions has less roots than the number of nonzero coefficients (the number of sign changes in the coefficient sequence, respectively). These axioms are satisfied by monomials with real exponents on each interval not containing \(0\), by exponential functions on \(\mathbb{R}\), but not by logarithm functions with different bases. A noticeable feature of the approach is that it requires only elementary tools from mathematical analysis. Equally simple is the proof of the fact that, for \(n\) pairwise distinct functions \(f_j\) from such a family and \(n\) distinct real numbers \(t_i\), the vectors \(\bigl( f_j(t_1), f_j(t_2),\ldots,f_j(t_n) \bigr)^\top\) (\(1\leq j\leq n\)) form a basis of \(\mathbb{R}^n\).
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 5976244
zbMATH Keywords: Descartes' rule of signs
zbMATH Keywords: fundamental theorem of algebra
zbMATH Keywords: number of zeroes
zbMATH Keywords: real function family
zbMATH Keywords: real polynomial
zbMATH Keywords: sum of exponential functions
zbMATH Keywords: sum of logarithm functions
MaRDI profile type: Publication
full work available at URL: j.amc.2011.05.107
OpenAlex ID: W2086253178
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DOI: 10.1016/J.AMC.2011.05.107
description: scientific article; zbMATH DE number 5976244