A short proof of Kontsevich's cluster conjecture

Arkady Berenstein; Vladimir Retakh

doi:10.1016/j.crma.2011.01.004

Combinatorics/Algebra

A short proof of Kontsevich's cluster conjecture

Presented by: Maxime Kontsevich

Arkady Berenstein ¹; Vladimir Retakh ²

¹ Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
² Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 119-122.

Abstract
Résumé

We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map $K_{r} : (x, y) \mapsto (x y x^{- 1}, (1 + y^{r}) x^{- 1})$ are given by noncommutative Laurent polynomials.

Nous proposons une démonstration élémentaire d'une conjoncture de Kontsevich qui affirme que l'itération de l'application non-commutative rationnelle $K_{r} : (x, y) \mapsto (x y x^{- 1}, (1 + y^{r}) x^{- 1})$ est donnée par des polynômes de Laurent non-commutatifs.

Received: 2010-11-18
Accepted: 2011-01-04
Published online: 2011-01-20

DOI: 10.1016/j.crma.2011.01.004

Author's affiliations:

Arkady Berenstein ¹; Vladimir Retakh ²

¹ Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
² Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

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Arkady Berenstein; Vladimir Retakh. A short proof of Kontsevich's cluster conjecture. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 119-122. doi : 10.1016/j.crma.2011.01.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.004/

References
Cited by

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[9] A. Usnich, Non-commutative Laurent phenomenon for two variables, preprint, , 2010. | arXiv

Cited by Sources:

^☆ The authors were supported in part by the NSF grant DMS #0800247.

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