The exponent for the Markoff–Hurwitz equations

Arthur Baragar

Vol. 182 (1998), No. 1, 1–21

DOI: 10.2140/pjm.1998.182.1

Abstract

In this paper, we study the Markoff-Hurwitz equations x₀² + ... + x_n² = ax₀⋯x_n. The variety V defined by this equation admits a group of automorphisms 𝒜≅ℤ∕2 ∗⋯ ∗ 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 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

exists and depends only on n. We give an effective algorithm for determining these exponents. For large n, this gives the asymptotic result

Milestones

Received: 18 April 1996

Revised: 12 November 1996

Published: 1 January 1998