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Markoff-Rosenberger triples in arithmetic progression


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Author: González Jiménez, Enrique
Tornero Sánchez, José María
Department: Universidad de Sevilla. Departamento de álgebra
Date: 2013-06
Published in: Journal of Symbolic Computation, 53, 53-63.
Document type: Article
Abstract: We study the solutions of the Rosenberg–Markoff equation ax2 + by2 + cz2 = 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.
Cite: González Jiménez, E. y Tornero Sánchez, J.M. (2013). Markoff-Rosenberger triples in arithmetic progression. Journal of Symbolic Computation, 53, 53-63.
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DOI: 10.1016/j.jsc.2012.11.003

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