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A shadow Markov equation
Nathan Bonin1; Valentin Ovsienko1 
1 Laboratoire de Mathématiques de Reims, UMR9008 CNRS, Université de Reims Champagne-Ardenne, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 Reims cedex 2, France
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1483-1489.

We introduce an analogue of the classical Markov equation that involves dual numbers a+αε with ε 2 =0. This equation characterizes the “shadow Markov numbers” recently considered by one of us. We show that this equation is characterized by invariance by cluster algebra mutations.

Received:
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Accepted:
Published online:
DOI: 10.5802/crmath.496

Nathan Bonin 1; Valentin Ovsienko 1

1 Laboratoire de Mathématiques de Reims, UMR9008 CNRS, Université de Reims Champagne-Ardenne, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 Reims cedex 2, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Nathan Bonin; Valentin Ovsienko. A shadow Markov equation. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1483-1489. doi : 10.5802/crmath.496. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.496/

[1] Charles H. Conley; Valentin Ovsienko Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group (2022) | arXiv

[2] Sergey Fomin; Andrei Zelevinsky Cluster algebras. I. Foundations, J. Am. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[3] Sergey Fomin; Andrei Zelevinsky The Laurent phenomenon, Adv. Appl. Math., Volume 28 (2002) no. 2, pp. 119-144 | DOI | MR | Zbl

[4] Andrew N. W. Hone Casting light on shadow Somos sequences (2021) | arXiv

[5] Yi Huang; Robert C. Penner; Anton M. Zeitlin Super McShane identity (2019) (to appear in J. Differ. Geom.) | arXiv

[6] Andreĭ Markov Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 15 (1879), pp. 381-406

[7] Gregg Musiker; Nicholas Ovenhouse; Sylvester W. Zhang Double Dimer Covers on Snake Graphs from Super Cluster Expansions (2021) | arXiv

[8] Valentin Ovsienko Shadow sequences of integers, from Fibonacci to Markov and back (2021) (to appear in Math. Intell.) | arXiv

[9] Valentin Ovsienko; Michael Shapiro Cluster algebras with Grassmann variables, Electron. Res. Announc. Math. Sci., Volume 26 (2019), pp. 1-15 | MR | Zbl

[10] Valentin Ovsienko; Serge Tabachnikov Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, Algebr. Represent. Theory, Volume 21 (2018) no. 5, pp. 1119-1132 | DOI | MR | Zbl

[11] James Propp The combinatorics of frieze patterns and Markoff numbers, Integers, Volume 20 (2020), A12, 38 pages | MR | Zbl

[12] Jeffrey M. Rabin Super elliptic curves, J. Geom. Phys., Volume 15 (1995) no. 3, pp. 252-280 | DOI | MR | Zbl

[13] N. J. A. Sloane The On-Line Encyclopedia of Integer Sequences, 2010 (http://oeis.org)

[14] Alexander Veselov Conway’s light on the shadow of Mordell (2022) | arXiv

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