The exponent for the Markov-Hurwitz equations
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 \[ \frac{\log n}{\log 2}<\alpha(n)< \frac{\log n}{\log 2}+o(n^{-.58}). \]
scientific article; zbMATH DE number 1060826