Markoff-Rosenberger triples in arithmetic progression

摘要

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.