On a conjecture of Posner and Rumsey
type: Item
instance of: scholarly article
title: On a conjecture of Posner and Rumsey
zbMATH Open document ID: 0813.11061
published in: Journal of Number Theory
publication date: 1995-06-07T00:00:00Z
review text: Let \(m_ 1 > m_ 2 > \cdots > m_{i-1} > 0\), \(a_ 1, \dots, a_ i\) be nonzero integers. An expression of the form \(\sum^{i-1}_{j=1} a_ j x^{m_ j} + a_ i\) is called standard \(i\)-nomial. \textit{E. C. Posner} and \textit{H. Rumsey} jun. [Mich. Math. J. 12, 339--348 (1965; Zbl 0144.03504)] conjectured, that if a polynomial with rational coefficients divides infinitely many standard \(i\)-nomials over \(\mathbb Q\), it divides a nonzero polynomial of degree less than \(i\) in \(x^ r\) over \(\mathbb Q\), for some \(r \geq 1\). Posner and Rumsey proved the conjecture in the case \(i = 3\). Let \(p\) be a nonzero polynomial with rational coefficients and \(k\) distinct roots. Denote by \(K\) the splitting field of \(p\), and \(d = [K : \mathbb Q]\). Let \(S\) be the set of places of \(K\) consisting of all infinite places and all valuations induced by the prime ideal factors of the nonzero roots of \(p(x)\) and set \(s = \text{card }S\). The authors prove the conjecture of Posner and Rumsey for \(i = 3\) in the following stronger quantitative sense: If \(p(x)\) divides more than \((4sd)^{s^ 6 \cdot 2^{180d} + 8sk}\) standard trinomials over \(\mathbb Q\), then it divides a linear or quadratic polynomial in \(x^ r\) over \(\mathbb Q\) for some integer \(r \geq 1\). Moreover, the authors show, that the conjecture is false for \(i \geq 4\). The main tool of the proof is the application of quantitative results on \(S\)-unit equations.
reviewed by: István Gaál
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 584045
zbMATH Keywords: divisibility
zbMATH Keywords: standard \(i\)-nomial
zbMATH Keywords: \(S\)-unit equations
zbMATH Keywords: polynomial with rational coefficients
author: Kálmán Győry
author: Andrzej Schinzel
MaRDI profile type: Publication
full work available at URL: jnth.1994.1026
OpenAlex ID: W2083170507
Wikidata QID: Q123263792
DOI: 10.1006/JNTH.1994.1026
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description: scientific article; zbMATH DE number 584045