On a class of Diophantine equations related to the numbers of cells in hyperplane arrangements
type: Item
instance of: scholarly article
title: On a class of Diophantine equations related to the numbers of cells in hyperplane arrangements
zbMATH Open document ID: 1215.11030
author: Ákos Pintér
published in: Journal of Number Theory
publication date: 2009-06-17T00:00:00Z
review text: The author studies the diophantine equation \[ {x\choose m}+\dots+{x\choose 1}={y\choose n}+\dots+{y\choose 1} \] in integers \(x\geq m\) and \(y\geq n\), where \(m\) and \(n\) are distinct integers greater than \(1\). The equation can be nicely interpreted geometrically: one asks when the numbers of cells in two simple hyperplane arrangements are equal. The author proves that for any fixed \(n>m\geq 3\) the above equation has only finitely many solutions. The main method of the proof is a deep theorem of Bilu and Tichy. Further, it is also proved that if \(m\geq 3\) and \(n=2\) then the soultions \(x,y\) can be effectively bounded in terms of \(m\). Here the main tool is a previous result of the author concerning the root structures of certain polynomials and the Baker-method. As it is pointed out, in fact the methods of the proofs are capable to handle much more general equations, as well.
reviewed by: Lajos Hajdu
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 5565744
zbMATH Keywords: binomial coefficients
zbMATH Keywords: polynomial Diophantine equations
zbMATH Keywords: cells is hyperplane arrangements
describes a project that uses: Maple
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.10.013
OpenAlex ID: W2154125790
cites work: Q5562698
cites work: The Diophantine equation f(x) = g(y)
cites work: The Magma algebra system. I: The user language
cites work: On \(S\)-integral solutions of the equation \(y^ m=f(x)\)
cites work: Q4345355
cites work: The product of consecutive integers is never a power
cites work: Q4348479
cites work: Q4530626
cites work: Q5582375
cites work: Q4293472
DOI: 10.1016/J.JNT.2008.10.013
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description: scientific article; zbMATH DE number 5565744