The exponent for the Markov-Hurwitz equations
type: Item
instance of: scholarly article
title: The exponent for the Markov-Hurwitz equations
zbMATH Open document ID: 0892.11009
author: Arthur Baragar
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: We study the Markoff-Hurwitz equations \(x_0^2+\dots+ x_n^2= ax_0\cdots x_n\). The variety \(V\) defined by this equation admits a group of automorphisms \({\mathcal A}\cong \mathbb{Z}/2*\cdots* \mathbb{Z}/2\) (an \(n+1\)-fold free product). For a solution \(P\) on this variety, we consider the number \(N_P(t)\) of points \(Q\) in the \({\mathcal A}\)-orbit of \(P\) with logarithmic height \(h(Q)\) less than \(t\). We show that if \(a\) is rational, and \(P\) is a non-trivial rational solution to this equation, then the limit \[ \lim_{t\to\infty} \frac{\log N_P(t)}{\log t}=\alpha(n) \] exists and depends only on \(n\). We give an effective algorithm for determining these exponents. For large \(n\), this gives the asymptotic result \[ \frac{\log n}{\log 2}<\alpha(n)< \frac{\log n}{\log 2}+o(n^{-.58}). \]
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 1060826
zbMATH Keywords: integer points
zbMATH Keywords: rational points
zbMATH Keywords: orbits
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: Markoff-Hurwitz equations
zbMATH Keywords: asymptotic result
MaRDI profile type: Publication
full work available at URL: pjm.1998.182.1
OpenAlex ID: W2048858343
DOI: 10.2140/PJM.1998.182.1
description: scientific article; zbMATH DE number 1060826