Markoff-Rosenberger triples in arithmetic progression

TL;DR

This paper investigates solutions in arithmetic progression for a generalized Markoff-Rosenberger equation over number fields, providing a decision algorithm and extensive computational results for specific cases.

Contribution

It introduces a complete decision algorithm for solutions in arithmetic progression and establishes finiteness results for these solutions over number fields.

Findings

Finiteness results for solutions in arithmetic progression

Decision algorithm for generalized Markoff equations

Computational analysis of specific cases over quadratic and arbitrary number fields

Abstract

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.