From Frobenius to Riedel: analysis of the solutions of the Markoff equation

Serge Perrine (1)

1 France Telecom - France Telecom

The beginning of the present article is a review of the preprint recently posted by Norbert Riedel on the site arXiv at Los Alamos, with the title "Markoff equation and nilpotent matrices", arXiv:0709.1499v4 [mathNT] 5 Aug 2008. The author of this document announced a proof of the Frobenius conjecture about the solution of the Markoff equation. The aim of the present work is to systematically verify all the Riedel's computations, to explain their significance, and to give some corrections. Our main task is to identify the tree dealt with by Riedel, to generalize his formalization to his whole tree, modifying when necessary his definitions. We clearly describe the mechanism devised by Riedel in order to prove the conjecture. Unfortunately, we put in light some deficiency in his developments, showing that the conjecture still resists. We also give explicit values for the parameters introduced by Riedel.

Mots clés: number theory; representation theory; Markoff equation

Domaines: Théorie des nombres [math.NT]; Théorie des représentations [math.RT]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Serge Perrine. From Frobenius to Riedel: analysis of the solutions of the Markoff equation. 2009. ⟨hal-00406601⟩