A combinatorial problem on polynomials and rational functions
type: Item
instance of: scholarly article
title: A combinatorial problem on polynomials and rational functions
zbMATH Open document ID: 0953.05005
author: Lajos Rónyai
author: György Elekes
published in: Journal of Combinatorial Theory. Series A
publication date: 2001-01-29T00:00:00Z
review text: Let \(C\) be a real number with \(C>1\). Let \(s\), \(t\) be two straight lines in the Euclidean plane, while \(U\), \(V\) are two collinear sets of \(n\) points each, located on \(s\), \(t\) respectively. Further, assume that among the \(n^2\) distances \(d(U_i,V_j)\) for \(U_i\in U\) and \(V_j\in V\), only some \(C_n\) are distinct. G. Purdy conjectured that if \(n> n_0(C)\), then the two lines must be parallel or orthogonal. This conjecture is related to a famous problem of \textit{P. Erdős} [Am. Math. Mon. 53, 248-250 (1946; Zbl 0060.34805)]. Using the characterization of restricted functions, the present authors settle this conjecture in the affirmative.
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 1421328
zbMATH Keywords: polynomial
zbMATH Keywords: rational function
reviewed by: Le Maohua
MaRDI profile type: Publication
full work available at URL: jcta.1999.2976
OpenAlex ID: W2045383484
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DOI: 10.1006/JCTA.1999.2976
description: scientific article; zbMATH DE number 1421328