An \(abcd\) theorem over function fields and applications
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.11031
author: Pietro Corvaja
author: Umberto Zannier
publication date: 2012-04-14T00:00:00Z
full work available at URL: smf bull 139 437-454.php
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6024209
zbMATH Keywords: abc conjecture
zbMATH Keywords: function fields
zbMATH Keywords: curves on algebraic surfaces
zbMATH Keywords: S-units
MaRDI profile type: Publication
OpenAlex ID: W153117969
cites work: Q5749404
cites work: Q4175362
cites work: Q5719099
cites work: Vanishing sums in function fields
cites work: Some cases of Vojta’s conjecture on integral points over function fields
cites work: Higher-dimensional algebraic geometry
cites work: A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS FOR PURE POWERS
cites work: Some remarks on the S-unit equation in function fields
DOI: 10.24033/BSMF.2613
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title: An \(abcd\) theorem over function fields and applications
published in: Bulletin de la Société Mathématique de France
review text: In the paper under review, the authors give a lower bound for the number of distinct zeros of the sum \(1+u+v\), where \(u\) and \(v\) are rational functions which is sharp when \(u\) and \(v\) have few distinct zeros and poles compared to their degrees. Their main result sharpens the ``\(abcd\)'' theorem of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results for Diophantine equations with polynomials. The main tool is a prior result of the authors from [``Some cases on Vojta's conjecture on integral points over finite fields'', J. Algebr. Geom. 17, No. 2, 295--333 (2008); addendum Asian J. Math. 14, No. 4, 581--584 (2010; Zbl 1221.11146)]. As applications, they obtain that the Fermat surface \(x^a+y^b+z^c=1\) contains only finitely many rational or elliptic curves when \(a\geq 10^4\) and \(c\geq 2\). They also obtain an interesting application to the so--called ``Diophantine \(k\)-tuples''. Namely, if \(a,~b,~c\) are three distinct nonzero complex polynomials not all constant such that \(1+ab=x^p,~1+ac=y^q\) and \(1+bc=z^r\) with complex polynomials \(x,y,z\) and integers \(p,q,r\geq 864\), then after suitably permuting \(a,b,c\), we have \(c^2+1=0\) and \(a+b=2c\).
reviewed by: Florian Luca
description: scientific article; zbMATH DE number 6024209