Generalized Perron Identity for broken lines
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 131-144.

Dans cet article, nous généralisons l’identité de Perron pour les minima de Markov. Nous exprimons les valeurs des formes quadratiques binaires à discriminant positif en termes des fractions continues associées aux lignes brisées passant par les points où les valeurs sont calculées.

In this paper, we generalize the Perron Identity for Markov minima. We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.

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DOI : https://doi.org/10.5802/jtnb.1071
Classification : 11J06,  11H55
Mots clés : Geometry of continued fractions, Perron Identity, binary quadratic indefinite form
@article{JTNB_2019__31_1_131_0,
     author = {Oleg Karpenkov and Matty van-Son},
     title = {Generalized {Perron} {Identity} for broken lines},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {131--144},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1071},
     zbl = {07246516},
     mrnumber = {3994722},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/}
}
Oleg Karpenkov; Matty van-Son. Generalized Perron Identity for broken lines. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 131-144. doi : 10.5802/jtnb.1071. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1071/

[1] Thomas W. Cusick; Mary E. Flahive The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, Volume 30, American Mathematical Society, 1989 | MR 1010419 | Zbl 0685.10023

[2] Harold Davenport On the product of three homogeneous linear forms. I, Proc. Lond. Math. Soc., Volume 13 (1938), pp. 139-145 | Article | MR 1574142 | Zbl 64.0146.03

[3] Harold Davenport On the product of three homogeneous linear forms. II, Proc. Lond. Math. Soc., Volume 44 (1938), pp. 412-431 | Article | MR 1575334

[4] Harold Davenport On the product of three homogeneous linear forms. III, Proc. Lond. Math. Soc., Volume 45 (1939), pp. 98-125 | Article | MR 1576815 | Zbl 0020.29304

[5] Vladimir V. Fock; Alexander B. Goncharov Dual Teichmüller and lamination spaces, Handbook of Teichmüller theory. Volume I (IRMA Lectures in Mathematics and Theoretical Physics) Volume 11, European Mathematical Society, 2007, pp. 647-684 | Article | Zbl 1162.32009

[6] Oleg Karpenkov Elementary notions of lattice trigonometry, Math. Scand., Volume 102 (2008) no. 2, pp. 161-205 | Article | MR 2437186 | Zbl 1155.11035

[7] Oleg Karpenkov On irrational lattice angles, Funct. Anal. Other Math., Volume 2 (2009) no. 2-4, pp. 221-239 | Article | MR 2506117 | Zbl 1246.11128

[8] Oleg Karpenkov On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators, Math. Notes, Volume 88 (2010) no. 1-2, pp. 28-38 (Russian version: Mat. Zametki 88 (2010), n° 1, p. 30–42) | Article | MR 2882161 | Zbl 1287.11087

[9] Oleg Karpenkov Continued fractions and the second kepler law, Manuscr. Math., Volume 134 (2011) no. 1-2, pp. 157-169 | Article | MR 2745258 | Zbl 05836521

[10] Oleg Karpenkov Geometry of Continued Fractions, Algorithms and Computation in Mathematics, Volume 26, Springer, 2013 | MR 3099298 | Zbl 1297.11002

[11] Svetlana Katok Continued fractions, hyperbolic geometry and quadratic forms, MASS selecta: teaching and learning advanced undergraduate mathematics, American Mathematical Society, 2003, pp. 121-160 | MR 2027174 | Zbl 1091.11002

[12] John Lewis; Don Zagier Period functions and the Selberg zeta function for the modular group, The mathematical beauty of physics (Saclay, 1996) (Advanced Series in Mathematical Physics) Volume 24, World Scientific, 1996, pp. 83-97 | Zbl 1058.11544

[13] Yuri I. Manin; Matilde Marcolli Continued fractions, modular symbols, and noncommutative geometry, Sel. Math., New Ser., Volume 8 (2002) no. 3, pp. 475-521 | Article | MR 1931172 | Zbl 1116.11033

[14] Andreĭ Markov Sur les formes quadratiques binaires indefinies. (second mémoire), Math. Ann., Volume 17 (1880), pp. 379-399

[15] Oskar Perron Über die Approximation irrationaler Zahlen durch rationale II, Heidelberger Akademie der Wissenschaften, 1921 | Zbl 48.0193.01

[16] Alfonso Sorrentino; Alexander P. Veselov Markov Numbers, Mather’s β function and stable norm (2017) (https://arxiv.org/abs/1707.03901) | Zbl 1431.37051