A $q$-analog of the Markoff injectivity conjecture holds - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

A $q$-analog of the Markoff injectivity conjecture holds

Résumé

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a long-standing open problem, is then equivalent to injectivity on Christoffel words. A $q$-analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this $q$-analog. The proof is based on the evaluation at $q = \exp(i\pi/3)$. Other roots of unity provide some information on the original problem, which corresponds to the case $q=1$. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.

Dates et versions

hal-04245628 , version 1 (17-10-2023)

Identifiants

Citer

Sébastien Labbé, Mélodie Lapointe, Wolfgang Steiner. A $q$-analog of the Markoff injectivity conjecture holds. 2023. ⟨hal-04245628⟩

Collections

CNRS ANR
13 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More