Markoff–Rosenberger triples in arithmetic progression

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.

On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

“…Of special interest is the connection of the Markoff equation with the classification and description of the quiver algebras with three vertices ( [11]). Several results concern with the solutions of an equation of type (MR) in the integers, in the rational numbers and, more generally, in algebraic number fields ( [2], [4], [5], [13], [18], [25], [29], [30], [33], [36], [37], [38], [40]) or with the asymptotic behavior and the growth of the integral solutions ( [3], [7], [8], [46]). Hence, we here do not especially look at the equations of type (MR).…”

Markoff–Rosenberger triples in geometric progression

“…has been deeply studied in [10,11,12,9]. Recently, the author and J. M. Tornero [22] have studied the case of triples in a.p. on the Markoff-Rosenberger equation over number fields.…”

“…to prove results for g.p. analogous to those obtained in [22]. That is, our main objective is to study the set GP (a,b,c,d) (K) := {O K -non-trivial triples in g.p.…”